Simple check for the global shape of the Earth I have been on a date recently, and everything went fine until the moment the girl has told me that the Earth is flat. After realizing she was not trolling me, and trying to provide her with a couple of suggestions why that may not be the case, I've faced arguments of the like "well, you have not been to the space yourself". 
That made me think of the following: I myself am certain that the Earth is ball-shaped, and I trust the school physics, but being a kind of a scientist, I could not help but agree with her that some of the arguments that I had in mind were taken by me for granted. Hence, I have asked myself - how can I prove to myself that the earth is indeed ball-shaped, as opposed to being a flat circle (around which the moon and the sun rotate in a convenient for this girl manner). 
Question: Ideally I want to have a proof that would not require travelling more than a couple of kilometers, but I am fine with using any convenient day (if e.g. we need to wait for some eclipse or a moon phase). For example, "jump an a plane and fly around the Earth" would not work for me, whereas "look at the moon what it is in phase X, and check the shape of the shade" would. 
Trick is, I know that it is rather easy to verify local curvature of the Earth by moving away from a tall object in the field/sitting on the beach and watching some big ship going to the horizon. However, to me that does not prove immediately that globally the Earth has same/similar curvature. For example, maybe it's just a shape of a hemisphere. So, I want to prove to myself that the Earth is ball-shaped globally, and I don't want to move much to do this. Help me, or tell me that this is not possible and why, please. As an example, most of the answers in this popular thread only focus on showing the local curvature.
P.S. I think, asking how to use physics to derive global characteristics of an object from observing things only locally (with the help of the Sun and the Moon, of course) is a valid question, but if something can be improved in it, feel free to tell me. Thanks.
Update: I have not expected such a great and strong feedback when asking this question, even though it is indeed different from the linked ones. Them are still very similar, which was not grasped by all those who replied. I will thoroughly go over all the answers to make sure which one fits the best, but in the meantime if you would like to contribute, please let me clarify a couple of things regarding this question: they were in the OP, but perhaps can be made more obvious.


*

*I do not have a goal of proving something to this date. I see that mentioning her might have been confusing. Yet, before this meeting I was certain about the shape of the earth - but her words (even though I think she's incorrect in her beliefs) made me realize that my certainty was based on the assumption I have not really questioned. So sitting on a beach with another friend of mine (both being ball-believers) we thought of a simple check to confirm our certainty, rather than to convince anyone else in us being right.

*I am only looking for the check that would confirm the GLOBAL shape of the earth being ball-like. There were several brilliant answers to another question that worked as a local curvature proof, and I am not interested in them. 

*I am looking for the the answer that will show that the Earth is ball-shaped (or rather an ellipsoid), not that it is not flat. There are many other great shapes being neither ball/ellipsoid nor flat. I do still have an assumption that this shape is convex, otherwise things can go too wild and e.g. projections on the Moon would not help us.
I think point 1. shows why is that a valid physics/astronomy question, rather than playing a devil's advocate defending the flat Earth hypothesis, and I would also happily accept the answer like you cannot show this by not moving for 20k kilometers because A, B, C if there's indeed no simple proof. At the same time, points 2 and 3 should distinguish this question from the linked ones. 
 A: So, I know math is never a good idea when dealing with this kind of people, but one local quantity that depends on the global properties of Earth is the gravitational field.
The gravitational potential of a sphere is $$V_S=-\frac{GM}{r}$$ in particular if you take $r=z+R$, where $R$ is the radius of the Earth a $z$ is the distance from Earth's surface (assuming Earth is a smooth ball essentially), you get
$$V_S\approx-\frac{GM}{R}+\frac{GM}{R^2}z$$ where $g_S=\frac{GM}{R^2}\approx 9.81m/s^2$ and the local gravitational field in the vicinity of Earth surface is $\vec{g}=(0,0,-g_S)$ as you learn in High school hopefully.
Now the gravitational potential of a disk is a bit more complicated, if you take the disk on the $x,y$ plane, then the potential on the $z$ axis orthogonal to the plane is
$$V_D(z)=\frac{2 GM}{R^2}\left(\frac{z}{\sqrt{z^2+R^2}}-1\right)$$ where $z$ is the distance from the center of the disk and $R$ is the radius of the disk and where we have assumed that the disk is homogeneous.
In the proximity of the disk, when $z\approx0$ we can approximate the potential to
$$V_D\approx\frac{2 GM}{R^2}\left(-1+\frac{z}{R}\right)$$
and so the local gravitational field would be $\vec{g}=(0,0,-g_D)$ where $g_D=\frac{2GM}{R^3}$, so for a disk the local gravitational acceleration is different than for a sphere.
Ok, so now suppose you know the radius of the flat Earth is the distance between the north and the south pole, namely $R_D=\pi R$ where $R=6.378\times 10^6 m$ is the radius of Earth (the true one...). Now, since you know from trivial observations you can make at home that $g=9.81 m/s^2$, then you can use $g_D$ and $g_S$ to estimate its mass if it was a disk or a sphere:
$$M_S\approx5.98\times 10^{24}KG\\
M_D\approx5.91\times 10^{32}KG$$
so for Earth to be a disk it would be more massive than the sun! Therefore you should rework the whole Copernican model of the Solar system to make this ordeal work.
And also bodies would fall rather differently if you throw them high enough.
A: The reality is that you will never prove a global effect to a skeptic.  It is simply not possible.  I'm a skeptic, so I'm very familiar with how you dance around the evidence.  It's kind of fun, really.  All I have to do is make sure I refuse to accept any macroscopic effect outside of my own line of sight!  Find a clever way to get it into my line of sight, and I'll demand that I can touch it.
It infuriates people who are used to giving science a monopoly on truth.
If you really want to convince a person like this, it will take time.  You have to tighten the noose slowly.  You cannot provide an argument which will disuade them, but they can.  Give them enough time to flesh out their opinion about how things work, and then start pointing out inconsistencies in them.  (Preferably poke holes in places where your round earth beliefs correctly match the evidence.)
The most immovable FE position is one which avoids ever making any predictive statements about what the Earth may look like, and merely challenges that your data is not 100% perfect.  Since no scientific data can ever possibly yield 100% certainty, they'll get to play that card forever until they are forced to pony up a model of their own.
Personally, I find the most useful approach is to look at airline travel.  If flat earth is accurate, then there will be a 2d shape for the earth which does a very good job of approximating the time of different flights.  IF round earth is accurate, then there will be a sphere (or almost a sphere) which does a much better job of predicting flight times.  Nothing will be exact, because planes fly at different speeds, but you should find that the flat earth model does a horrendous job of predicting plane flight times.
I like this approach because it doesn't require leaving your keyboard, if you don't want to.
The flat-earthers I've argued this point with typically turn conspiracy theorist at this point.  If every commercial pilot ever is in on a great conspiracy to fool us into believing there's a round earth, along with every astronaut, and everyone who has to design their equipment, then flat-earth remains viable.  There's a lot of money in the airline industry, so taking funny flightpaths to obscure the flatness of the earth would be a costly business move, which deepens how deep and thorough that conspiracy has to be to work.  Thus, they aren't necessarily dissuaded, but as I said before, this is a long process.  You won't convince someone overnight.
But you can see what sort of person you're dating.  It can get really amusing when their theory requires the conspiracy and utter perfect secrecy of 5% of the world population.
Oh, and whatever you do, never make the mistake of suggesting the earth is as sphere.  The best evidence we have is that it's a rather rugged uneven shape with all these things called "mountains" that don't appear on any self respecting sphere.  Any good FE will make you regret that simplistic phrasing.  You can say that the evidence suggests that it is reasonable to say the WGS84 geoid is a pretty bloody good approximation.  Certainly substantially above par of any flat earth argument I've been given yet.
Though nobody has been able to shake my confidence that the earth actually rests on the back of a giant turtle.  And that turtle stands on the back of another turtle.
It's turtles all the way down.

I love conspiracy theories.  It's comforting to think that someone is in that much control of things.

A: Most answers so far have focused on geometric/optical arguments. There is also an important dynamical argument. 


*

*The global shape with lowest gravitational potential energy is a spherical ball, cf. e.g. this Phys.SE. 

*When we add the centrifugal$^1$ potential energy, it becomes an oblate spheroidal shape, cf. e.g. this Phys.SE post. 

*Now how much can Earth's shape deviate from the above hydrostatic equilibrium? In other words, what is theoretically the height $h$ of the tallest mountains/deepest valleys possible? Given that the materials of the Earth can stand stresses up to, say, $\sigma \sim 10^3 \text{ atm}$, a back-of-the-envelope-calculation suggests an order-of-magnitude $h\sim \frac{\sigma}{\rho g}\sim  10\text{ km}$, cf. e.g. this Phys.SE post. This rough estimate is comparable with the actual mountains on Earth. 

*Needless to say that a flat disk-shaped Earth is incompatible with the above dynamical argument.

$^1$ We assume that |centrifugal force| $\ll$ |gravitational force|. For the opposite extreme, check out this Phys.SE post.  
A: This may not qualify as a direct observation, but you can indirectly measure distances between distant cities. According to this map from the Flat Earth Society the distance between Australia and Argentina should be much larger than between, say, Alaska and Ukraine. These days, there are ways to check that distance without leaving your computer.

Measurement 1: You can go to the website of your favourite travel agent and look at flight times to verify that distances more or less match the hypothesis of a spherical earth. You might also consider the limited range of commercial jet aircraft that would make distances e.g. New Zealand - Chile impossible on a flat earth map. Australia - Argentina should pass over the Arctic. For a rigorous confirmation, you can of course take one of these flights.
Measurement 2: ping times and traceroutes will roughly correspond to the distances between server nodes. You'll have to allow for the paths the connections take, which may not be the shortest distance between two points, but can be found out. There will also be some delays within routers/switches etc. I haven't done the experiment myself, but I imagine that the results would be much closer to what the spherical earth model predicts than what the flat earth model predicts. Once again, the difference should be particularly large when you consider Northern countries vs Southern countries. You can use a site like https://asm.ca.com/en/ping.php to ping to/from lots of places quickly.
A: Look at the Moon during a lunar eclipse. Or during any other Moon phase for that matter, where part of the Moon is shaded.
As ironically exemplified by Neil deGrasse Tyson in a Twitter tweet:

Neil deGrasse Tyson @neiltyson
November 26, 2017
A Lunar Eclipse flat-Earther's have never seen.


We always only see a circular shade as if the Earth really is a sphere. Sure, it is possible even with a flat Earth that the arrangement of Sun-Earth-Moon just happens to give a circular shade by coincidence. But then we would only sometimes see a circular shade - other times we should see something like the above image.
Just wait for the next lunar ecplise or Moon phase. Or for the next 100 ones. Surely, at some point you must be seeing the shade from another angle giving an elliptic shade. Or a thin shape as the picture.
These observations can be done with our own naked eyes. And we have never, ever observed any sign of a flat-Earth shadow.
A: This is really cool because I have also quite recently came upon this question, only without the "local curvuture" catch. Al Nejati's answer is already quite extensive, but I thought his first explanation was particularly interesting. Then I started thinking about the possible counter arguments. Why can't you see distant tall object if the Earth is flat? 
Maybe your eye sight is the limit. You can only see a certain distance away before things begin to get blurry. But then, let's take an unobstructed stretch of land, you on one side, and a tall object a certain distance away, beyond your field of vision. 
Assuming clear weather, if you approach the object until it's in your field of vision if the Earth is flat then you will always see the bottom of the object first, because it's the closest point to you. However, as we all know when you approach a tall object you never see the bottom first, usually the top. In any case it will be some point between the top and the bottom assuming the object is unobstructed. Since there aren't any examples of the opposite happening the Earth can't be flat. 
A: Great skeptic question - although I do have to mention that of course we can not have evidence and knowledge of everything: sometimes we must just trust experts (!
I like some of the answers above and I admit I did not read ALL of them, so I apologize for any duplication. However I will try to address it from another point of view. I realize my proposal is unrealistic in real life and is not really what you are asking for, but I still think it is worth mentioning it as it can lead to some results, if the other party is open minded enough.
A brief intro: the following can also be conceived as a thought experiments, it does not require actual measurements. However, it is in principle possible, and for many of the required tasks easy, to do an experimental/mathematical verification, as you ask.
First of all, I think something all parties should agree with, at the beginning, with a set of assumptions: do we trust classical mechanics in general? If yes, jump the next sentence. 
If not, rederive $F=Ma$ from either thought experiments or a set of inclines measurement with decreasing friction etc. etc.
If the other party refuses this at all, well, then there is little you will agree on later on, except first hand experience, which can lead to basically any argument and counter-argument.
Assuming classical mechanics hold and Newton's law of gravitation works, show her that to explain how object falls we need a model of Earth, of course you propose a spherical one and let him/her propose their own. Make a prediction about the acceleration of gravity or any other relevant quantity and test the two models.
The other party will probably have a wrong prediction. If it is so, no problem: let her/him find a model which works. As soon as the prediction works, let's compute another quantity or make another prediction. For example, if she finds a way to show that a flat disk will have the same local acceleration of gravity, then ask her: ok, what about the borders of the disk? Does it work there too? Should not we see a difference depending on how gar from the center we are? Ask her/him to refine the model until this other prediction (perfectly predicted by the spherical Earth model, by the way) is also corroborated.
Go on, until she is convinced or she finds a model which predicts everything. If the latter is the case, I am afraid her model has to be pretty valid. It will probably be a sphere tho. 
The point is that many of the experiments suggested above work but always have some caveats. However, taken all together they acquire bigger value.
The big flaw here is that this could last forever in principle and it is an over-simplification of how the scientific method works. The other party may complain that she does not have a physical background, that somehow somewhere you are fooling her. You can then let her ask for help from a friend, google, do a PhD in Physics - the point is always that in principle, unless she rejects major points like multiplication or the existence of physics laws, the argument can be settled.
This is an oversimplification of the scientific method. Experiments, theory, first hand experience can all be joined in the quest for the truth. Of course after a while computation becomes cumbersome, but that may even help realize why we trust experts: they did this work for us and showed us how to in principle repeat it. It does not mean they are right: it means if they are wrong this can be figured out.
The big point is that you work together, assuming a fixed set of axioms, to uncover the model which best fits all data (this does not mean the model is true). Eventually, you may even realize that Earth is not really a sphere (it depends on the frame of reference at relativistic speeds and/or is actually a wave function :D ).
As a closing sentence, I want to stress that in a time like this, of big disbelief and aggressiveness, it is especially us, the scientists, the skeptics, that have to call for dialogue and sharing, instead of just labeling the other parties as minus habens: even if they refuse everything, try to question their refusal, discuss, convince, show evidence. Of course there will always be the dumb one or the malevolent one and so on - but the majority of the people should be handled this way, I think.
A: Launch a weather balloon with a GoPro attached.
Not everyone has a weather balloon to hand, but they are readily available to people who use them.  Talk to your local university about who their meteorological researchers are.  The Earth's curvature will be visible from a high-altitude weather balloon.
The Register's LOHAN project released a plane from a weather balloon.  Students at the University of Leicester have also sent up a high-altitude weather balloon.  Of course you need to allow for the camera lens - the LOHAN camera shows a clear "fisheye" effect, but that is something your date could confirm for herself.  Once she's confirmed that the camera provides a true representation of straight lines, or at least analysed where it does not, then she has to accept its results.  Either that, or she has to come up with her own alternative hypothesis for why some kind of lensing would occur.
This isn't totally straightforward, but is certainly within the realms of "citizen scientists".
A: Get something like a GoPro (or better a more suitable camera with less fisheye effect. Thanks to Åsmund), a GPS tracker and a helium balloon. 


*

*Attach camera and GPS tracker to the balloon

*let the balloon go off high in the air

*just collect the SD card after the balloon came down again (track location with GPS)

*watch video with the girl and prove that earth is not flat


(You need to get the right size of the balloon and the correct amount of helium to get the balloon high enough. And you need to attach the camera in the right angle and fix it there. Also find a suitable place to let the balloon of. Maybe there are some legal requirements too but who cares)  
There are some videos on YouTube from people who have done that. (But since I don't like YouTube, I won't link them here)   But here is a nice link about that topic Physics students from the University of Leicester have captured breathtaking images of the Earth's stratosphere using a high altitude weather balloon.
A: I think one phenomenon that is hard to refute using the flat earth theory is: Star Trails.


*

*Capture the star trails in Northern Hemisphere in a time lapse video and you'd notice that the stars move in an anti-clockwise direction. Here's a video 

*Repeat the same experiment in the Southern Hemisphere and you'll notice the Star Trails moving in a clockwise direction.
Once these videos are obtained, you could demonstrate and explain this phenomenon using a soccer ball, or a basketball, a camera that can be mounted on it, and a picture of stars printed on, say, an A3 paper. 


*

*Mount the camera on top of the ball, hold it under the picture of stars, and rotate it in an anticlockwise direction (as the earth would). The video thus captured would corroborate with the movement of the stars as observed in the time lapse video from the Northern Hemisphere. 

*Likewise, attaching the camera at the bottom of the ball, placing the picture on the floor, and maintaining the same direction of rotation would show the stars taking a clockwise direction.
A: Great, she has a hypothesis. Make her prove her point.
You: What evidence do you have that it's flat?
Her: "Because it looks flat."
You: "What diameter would a round Earth have to be before it would look flat to you?"
Make her do the math. If she can't do the math then she has no basis for making any conclusions herself and she simply chooses whom to believe.
In that case, why does she believe those few people over everyone else? And, why would everyone else lie?
Also, "have you been to space?" Would she believe someone who actually did go to space or is that part of the conspiracy?  Where do the rockets go that we see?  Is everyone who goes up in a rocket in on the conspiracy? And why is there a conspiracy in the first place? What is the reason to lie about if it is round or flat? We've had people from dozens of countries around the world go into space.  Are all the countries in on this or is it just the US?
If she can't open her mind to the reality before her, you're better off cutting her loose now because this type of thinking applies to so many other areas of life and can be a problem when it comes to vaccines for children, etc., etc.
This kind of stuff just makes me sad.
A: Time Zones
If you or your date know even one person living several time zones away that you trust, call them with a video call and let them tell you and show you that it is, say, night when on your side it is day. Then turn the tables and ask the flat-earther to explain this.
Of course, flat earthers have explanations for this, for example this one: https://imgur.com/gallery/eyuUVdc
How this explains sunrise and sunset, I have not the faintest idea, but I'm sure the flat earther will have no trouble explaining. Right?
Now add the death blow:
Latitude and Seasons
If you or your date know at least one person living at a considerably different latitude, call them next on video and ask them about the angle of sunlight. Now let the flat earther explain the different angles of sunlight, and check that this explanation does not collide with the one above.
If their model isn't dead yet, check how it explains seasons. If it does, the model is certainly so convoluted, that just explaining it should make the person stop in their tracks and realize just what nonsense it is, and then you can have a laugh together. 
Why this works
These are easily observed everyday facts that don't require any special knowledge to see or understand. The only challenge possible to them is that someone is lying, which is why you need a trusted person on video, so you have both their word and your own eyes of a live event. To both fake the video and make a trusted person lie, both in real time, with no preparation time, requires a mind-boggling conspiracy.
Most flat earth theories can easily explain away each individual evidence of a spherical earth. But their explaning-away ideas often collide with each other. The day-and-night "explanation" above does not explain sunrise, sunset, latitude effects, solar or lunar eclipses, etc.
What you are trying to do
You are not trying to convince anyone. You show (or fake) interest in their position and ask them to explain facts that you both agree upon (the sun indeed rises and sets, etc.). If you do this rigorously, they by necessity arrive at the same conclusion, because this is the conclusion that fits all the observed facts.
What you should do instead
Don't date lunatics. ;-)
(I have expanded this into an answer, but credit for the idea goes to @Al Nejati.)
A: Ask her what happens when you get to an edge, then offer to take a cruise or a flight or a drive with her to get there. Where is her idea of where the edge is?
Also, gravity would change as you get to an edge and away from the center of the flatness, yes?
Update 10/24/18: this just occurred to me after reading a topin on line of sight for recieving some kinds of radio data.  Take a tightly focused laser, point it parallel with the ground, and start walking away from it. If the Earth was flat, the laser distance above the ground would never change. If the Earth is round then after not very many miles the laser beam level would start increasing in height.
A: Simple proof is at an equinox - a day of equal day and night ie 12 hours of each wherever you are on the earth.
Barring large geographic barriers, ie mountains or valleys, it can easily be observed that no matter the location of the observer on the equinox at 6am local geographic time the sun will rise due East.  Twelve hours later at 6pm local geographic time it will set due West.  On the equator it will pass directly overhead at local noon.
On a flat earth with the small sun 'orbiting' the centre of the disc then the sun will only be on the direct east-west line at the equator at local noon and will spend the rest of the day to the north or 'hubward'.  At any time on a flat earth all those along a single line of longitude, a straight line from the 'hub' to the 'edge' or 'wall', would observer the sun at a different bearing, even during sunrise and sunset on an equinox, and that goes against actual observations made.
The Earth is not flat - QED
A: Since the world-wide web is actually world-wide, it should be very easy to prove that the earth is not flat, and relatively easy to prove that the earth actually is round.  To do this, send out a message to the world, looking for respondents in various countries.  Assuming that enough people reply, make sure that you get respondents that are widely scattered, both in latitude and longitude.  Then, at a previously agreed upon date, send out a message asking everyone to determine the angle of the sun in the sky at that moment, in terms of both altitude and azimuth.  Have all those people report those measurements back to you.  Then, on a small globe, set up a demonstration where a flashlight takes the role of the sun, and show that for a very particular orientation of the flashlight and locations on the globe, all the measurements match those same measurements in the demonstration.
A: I love this question, because it's a very simple demonstration of how to do science. While it's true that in science one should never accept anything 'without evidence', it's also true that blind skepticism of everything and anything gets one nowhere - skepticism has to be combined with rational inquiry. Your date has gotten the 'skepticism' part of science, but she's failed to grasp the equally-crucial part where one looks at the evidence and thinks about what the evidence implies. You cannot just refuse to think or accept evidence. If your goal is to learn nothing, then nothing is what you'll learn.
There are many, many ways of verifying that the Earth is not flat, and most of them are easy to think about and verify. You certainly do not need to go to space to realize the Earth is round!
If the Earth is flat, why can't you see Mt. Kilimanjaro from your house?
Mt. Kilimanjaro is tall, probably taller than anything in your immediate neighborhood (unless you live in a very deep valley) and so the question is why wouldn't you be able to see it from anywhere on Earth? Or, for that matter, why can't you see it from even closer? You have to be really close, in planetary terms, to be able to see it. This wouldn't be true if the Earth were flat!
One might argue that this is just because of the scattering of the atmosphere. Distant objects appear paler, so probably after some distance you can't see anything at all.
So then let's think about things that are closer. Stand on the ground and the horizon appears only a few km away. Go to the top of a hill, or a large tower, and suddenly you can see things much farther away. Why is this the case if the Earth is flat? Why would your height above ground have anything to do with it? If I raise or lower my eyes with respect to a flat table, I can still see everything on that table. The 'horizon' of the table never appears closer.
If the Earth is flat, why do time zones exist?
Hopefully your date realizes that time zones exist. If not, it's pretty easy to verify by doing a video call with someone in a distant location. The reason for time zones, of course, is that the sun sets and rises at different times at different parts of the globe. Why would this be the case? On a flat Earth, the sun would rise and set at the same time everywhere.
If the Earth is flat, why is the Moon round?
The moon is round and not a flat disc, as you can see by the librations of the moon. What makes the Earth special, then?
Further, all the planets are round, although to verify this you need a good telescope. Again, what makes the Earth special?
If the Earth is flat, then what is on its 'underside'?
Hanging dirt and leaves? A large tree? Turtles? Those who reject the roundness of Earth either have no explanation or their explanation is based on much less solid grounding than the pro-round arguments (which, of course, is because the Earth is not flat).
If there is 'nothing' under the Earth, then lunar eclipses would make no sense as the Earth needs to be between the Moon and the Sun.
EDIT: As to the question of whether the Earth is round or some weird hemisphere/pear/donut shape, among other things those would all lead to a situation where gravity is wrong. For a hemisphere for example, gravity would not point down (towards the Earth) at any point on the Earth's surface unless if you were sitting right at the top of the hemisphere. Similar arguments can be made for the other shapes.
Sure, it's possible to make it 'work' by doing even stranger things like altering the distribution of mass and so on, but at that point you've gone very far into violating Occam's razor.
A: You could draw her a picture like this:

Imagine being on the beach and looking at the horizon. In the case of a curved earth you will see a sharp transition between sea and air, since everything below the dashed line will be visible as sea while everything above the dashed line is visible as air.
In the case of a flat earth you could in theory see infinitely far. This would mean anything that's high enough is in your line of sight, including objects across the ocean. Ofcourse sight will get worse over large distances due to atmospheric scattering but that would mean you would either

*

*See anything that's in your line of sight

*See a blur instead of a horizon.

Anyone has seen a beach either in real life or on pictures, so this should give some intuitive evidence for curved earth. If you know the distance between you and something on the horizon you can even estimate the radius of the earth.
Note: these pictures are not drawn to scale but that should be obvious
A: You can pin down the global shape of the earth pretty well by considering how different people over the globe will see reference objects, like the stars and the sun.
The Stars
Most maps of the flat earth place the north pole in the center, and stretch the south pole out at the circumference. Imagine that we had three people standing at three, far away, southern locations in Africa, Australia, and South America. Say that when night falls in each region, the observers look south. What do they see?
 
In real life, they will all see the southern celestial pole. They will all see the same stars moving around a singular point somewhere in the sky. (As you go more and more south, this point rises higher and higher in the sky.)

This is because, as the observers are actually on a sphere all looking in the same direction, they are all looking at the same stars. As the earth rotates through the night, they will see the effects of this rotation because the stars will all be moving around a fixed point in the sky.

You can also observe such timelapse footage in this video.
Sometimes flat earthers say something about the "refraction of light" causing this. For one, they are unable to experimentally demonstrate how light could ever "refract" in the way they need. More severely, there is actually no way that light could refract to give the same celestial pole, with the same stars observed, to all three observers.
The Size and Position of the Sun
As objects get closer, they look bigger. In the round earth planetary model, because distance from the sun to the earth is much larger than the diameter of the earth, the sun will look the same size to all people on earth who can see it.
This presents another problem for the flat earth model. Imagine two scenarios, one where the sun is pretty close, and one where it is very far away.

Most flat earthers believe in a model more like the one on the left. The problem with this model is obvious. People who see the earth from different vantage points will see the sun looking drastically different sizes. However, if we place the sun very far away, we see a much worse problem: the sun will always be directly overhead for everybody. It will never get even close to the horizon.
In the flat earth model, there is always an unresolvable tension between the apparent size of the sun and the apparent position of the sun in the sky for all observers.
Actually, neither the model on the right or the left can reproduce a sunset (or explain why the sun only lights up half of the earth at a time.)
A: There are some simple experiments that you can do to demonstrate the shape of the Earth.  You can watch things disappear over the horizon as they move away from you - ships being the standard example.  You can drive toward a tall building or a mountain across a nice level plain and watch as it comes into view top-first.
If you're in the West you can go to the beach and lay down near the water, watch the sun set, then stand up and see it set again.  In the East you can do something similar, but watch the sun rise while standing then lay down and watch it appear again.  The differences in both cases are quite small so as soon as you see the first sliver appear in the East or the last bit disappear in the West then change your viewing height.
If you live anywhere near New Orleans you can go to Lake Ponchartrain and see the curvature of the causeway or go a bit West on I-10 and check out the power transmission lines.  Take some binoculars or a small telescope to get the best view of the curvature in the distance.
If you're close to the Equator then you can spend some time in the evening watching the Northern and Southern stars circling their respective celestial poles.  Or you can watch the satellites transiting the moon - sites like TransitFinder can help with this.
I could go on and on.  There are many many ways you can test this.
But before you try any of them I strongly recommend that you first take the time to find out how deeply into Flat Earth your friend is.  If she is watching all the normal Flat Earth videos on YouTube then she almost certainly already has trite answers for every observable phenomenon that you could possibly show her.  If she's a True Convert to this pernicious little meme then there is likely nothing you can do to change her mind.
The harsh reality is that there are many people out there who will believe in the face of all evidence to the contrary.  Nothing you can do will change them.
A: No matter which evidence you bring up, it can — and routinely will be by a Flat-Earther — refuted using a counter-argument along the lines of "You believe so because that's what you were taught". Unless you want to derive all of geometry, carry out all physical experiments through the history of mankind, etc., in front of the FE'er, every argument you can bring up ultimately hinges on your faith in other people's mathematical derivations and physical experiments, which, in the eyes of the FE'er, are flawed and/or deliberately altered.
Take for instance all the arguments given in the other excellent answers:


*

*The Earth's shadow on the Moon during a lunar eclipse would, at least sometimes, show any non-roundness of the Earth


*

*Lunar eclipses are caused by a satellite called "the Shadow Object".


*If the Earth is flat, why can't you see Mt. Kilimanjaro from your
house? (and many similar arguments like "Why do you see the mast of a ship before its hull?", and even "Why isn't it day all the time?")


*

*Even a on flat plane, parallel lines do not extent to infinity. What is even infinity? Infinity doesn't exist. The effect is caused by the angular limits of perception.


*If the Earth is flat, why do time zones exist?


*

*They match the position of the Sun's circular path above the disk.


*If the Earth is flat, why is the Moon round?


*

*The Moon is a planet, like Mars, Jupiter, etc. The Earth is not a planet.


*If the Earth is flat, then what is on its 'underside'?


*

*I don't know, and neither do you.


*A camera attached to a balloon shows the curvature of Earth


*

*The curvature is caused by a fish-eye effect. Or just photoshopped.


*Stars travel opposite ways on the northern and southern hemispheres


*

*The stars move in mysterious ways.


*If the Sun is so close to us, it should have a smaller angular diameter, the farther away it moves from us


*

*Magnification caused by the intense rays of light passing through the strata of the atmolayer (yup).



The reason of this is, in my experience, that FE'ers fundamentally do not understand how science works. They don't distinguish between laws, theories, hypotheses, facts, and ideas, they refuse to acknowledge helpful concepts such as Occam's razor and Russell's teapot, they cherry pick whatever experiment remotely seems to confirm their idea, and they disregard all evidence not confirming their idea as flawed/faked.
I realized this last year, when I participated in a radio program together with a FE'er. We went to the beach with his $87\times$ zoom camera, in an attempt to see the shore 19 km away. If he were right, we should be able to see the shore, and if I were right, we shouldn't be able to see the lower roughly 15 meters of the buildings there, disregarding atmospheric refraction. I was prepared that in fact we should probably see more of the buildings due to refraction (and was a little afraid that we would actually see the shore, since this was in the spring where refraction in warm air over cold water may be higher). It turned out we couldn't see the lower roughly 7 meters. Rather than taking this as evidence that at least the Earth isn't flat, he held it against me that my theory obviously is wrong. When I asked for at least some mechanism that might produce a flat Earth, he merely claimed that I also couldn't explain what might produce a round Earth. When I mentioned "gravity", he virtually quoted Phoebe from Friends: "Don't get me started on gravity".
I think the best you can do, if your date/friend/skeptic cousin is not already a completely convinced FE'er, to say that,

If indeed the Earth is flat, then either
  
  
*
  
*All physicists of the world are in some evil plot against humanity.
  
  
*
  
*This would imply that, whoever is behind this, would have means to make us all keep our mouths shut. There are roughly $\sim10^6$ physicists in the world today (Day 2015), and including past physicists increases this number to several millions. 
  
  
*All physicists are incompetent.
  
  
*
  
*This would imply that, so far only pure luck has held up airplanes in the sky, made communication possible through small, hand-held devices, enabled visually impaired people to see normally through metal-embraced pieces of glass in front of their eyes, etc. etc.
  
  

There is no proof that neither of the above is the case. But you could turn it around and claim that either 1) all members of the Flat Earth Society know that the Earth is round, but have some secret reason to keep this knowledge hidden from the public (implying that all $\sim10^3$ members$^\dagger$ of the FES keep their mouth shut), or 2) all members of the FES are incompetent (implying that… well, I don't really know what this implies).
How to prove to yourself that Earth is globally round
As in all physics, there is no proof. Physics doesn't prove anything, it verifies hypotheses continuously through experiments — preferably in several, mutually independent ways — until at some point we believe that a hypothesis describes something true, at which point we may call it a theory (or a part of a greater theory).
There are many ways to verify that Earth is round, both locally and  globally, but I think they all require you to either believe that other people are not a part of a plot (e.g. you'd have to believe that pictures of Earth taken from space are not faked), or to travel a bit (e.g. go to several locations and look at ships/bridges/land disappear below the horizon).
I don't see a way that, staying in the same place and only relying on your own eyes, you can check that the Earth is globally round. But if you will allow to contact friends in different locations, you can of course compare, e.g. which way stellar constellations are rotated, or the angle between the Sun and the horizon.

$^\dagger$I couldn't find any present numbers, but The Flat Earth Society had 189 members in 2010, according the their vice president.
A: If you were dedicated enough, you could travel all over the world, and conduct the experiment of Eratosthenes. He measured the circumference of the Earth with impressive accuracy by measuring the angle difference between parallel light rays from the sun at different geographic locations in Egypt. See here, for instance.
Now, you'll find that the curvature is roughly the same everywhere, and geometrically speaking, the only shape that has the same (nonzero) curvature everywhere is the sphere (and a hyperboloid but that would not be a serious consideration).
If we assume that the Earth is flat, then there's no way to explain the measurements. The assumption that the Earth is spherical does explain the data.
Furthermore, how can a flat earth have a circumference? The fact that we can measure the Earth's circumference seems to disprove a flat earth, especially in a global sense since the circumference is the same everywhere.
It's worth noting that there will be deviations above the average measured value along the equator, since the Earth is rotating (or is that just another conspiracy?).
A: I've been wondering the same thing. Using only items available in an average city, say what you can buy from Home Depot, Radio Shack, or find in a the labs of a typical highschool or community college, can you tell the world is round? How much can you tell without far away communication? 
Regarding the first question, I had a couple of ideas. 
They are basically all based on finding local tangent lines to the surface one is on, whether flat or round. 
Generalized Sun Dial
Take a block of wood and a nail. Record the length of the nail above the block after nailing it into the block of wood. Use a compass to Map off North. Mark off the shadow it makes at half hour intervals recording the angle it makes with North with a protractor, and the length of the shadow from the hole the nail makes in the wood. Also record the angle the sun makes above the horizon and its angular position clockwise from north. In particular take readings at Sunrise, Sunset, and Noon. Do this at multiple locations, varying latitude, longitude and elevation. Stay within the 25km range specified above. 
If you graph the Noon coordinates over the course of a year, you'd get roughly a figure eight suggesting either complicated motion of the earth, or of the sun. 
If you can accurately record sufficiently small deviations in Sunset and Sunrise times, curve fitting on Day Duration as a function of latitude  would vary by about 4.38 mins per degree of latitude at roughly the latitude of the mid United States. That range, 25km, allows for about 1.57 minutes of arc. So you'd need a fair amount of sensitivity, maybe milliseconds or so.
These measurements give you information about local, on earth effects, and effects in the Earth Sun System. For example, a Fourier analysis on the lateral motion of the sun in the recorded data will show a significant frequency corresponding to a cycle per 24 hours. There should be another detectable frequency at about 365.26 days. The latitude variation in Day Duration gives some information of the curvature of the earth, giving you shape information.
Curve fitting on the position of the sun should suggest the sun take on positions "below" the Earth. 
The variations in the length of the Nail's shadow and its angular position also effectively contains data about latitude. Length of shadow might also give information a bout time of year. 
Telescope
Determine the elevation and distance between multiple elevated objects in your 25km range. On a flat surface, the distance between the tops of the objects can be determined by the pythagorean theorem. Deviations from that measurement with position can give you information about the local geometry. For example, to first order, the deviation is proportional to the average of the elevations of point of observation and what is being observed, and inversely proportional to the radius of curvature, an infinite radius corresponding to flat surface. A telescope powerful enough to observe the craters on teh moon would be sufficiently strong to allow measuring of distances. The variation of how far you can see varying with height effectively gives you the tangent lines of the local surface. Tangent lines describe the local geometry. 
Sunlight Attenuation with  Altitude, longitude, and latitude
Sunlight Attenuation gives information of sun light traveling through the atmosphere with special behavior when the sun is near the horizon. Previously recorded information about the sun's position can be used to tease out attenuation effects due to the sun's proximity to the horizon vs what part of the horizon it sets at. The attenuation gradient should give geometric information of the surface. Atmospheric effects would have to be assessed.
This information should make a good case that one's local surface of the earth is shaped like a sphere. Just because something is locally a sphere, does it mean the rest of the planet is a sphere? 
Survey Methods
The previously mentioned methods should give a gobal  confirmation of spheroidal earth. 
Take a random, sufficiently representative sampling of people all over the world via phone. One finds that 50% of the world is illuminated by the sun at any given time. Mapped out, the Day/Night boundary forms a sinusoidal curve with shape and size varying in a particular way. A similar sinusoidal curve would appear on a disk model. The longitude and latitude of the observation points combined with the previously described measurements should give global geometric evidence not only against a flat earth but for a specific shape. 
A: There are so many ways to demonstrate this. You could stand on the shore and watch a sail boat sailing away from shore. You will see that as it passes over the horizon it disappears from view from the bottom up to the top of the mast. You can lie on a beach and watch the sunset. If as soon as it disappears over the horizon you stand up, it’s possible be able to glimpse it again. 
These concepts are not particularly new or modern. The fact of the curvature of the earth was known well before modern science or math. The concept was also why ancient sailing ships had a “crows nest” atop the mast for positioning of look outs to see enemy ships or land earlier than they would otherwise have done from on deck.
A proof based on math would be to sketch out a triangle. On a curved surface the three angles of all triangles add up to more than 180 degrees which is not possible on a flat surface.
A: For a flat disk, the local gravity field is different around the edge than the center (say, the field line is not vertical), and very different to everywhere on a sphere. 
Hence "flatland theory" predicts: there exist places at which you drop a ball (or simply look at the trees), then the ball will always "fly" away, and then land. However, such a phenomena has never been found. 
A: Make a map with a drone
So, what you'll need to do is get a drone, and just map out the entire Earth. Now you can calculate the Gaussian curvature of each point. Now you can figure out the shape of the Earth:


*

*If the curvature is approximately $0$ (averaged over large areas), the Earth is flat. Guess NASA was lying to us.

*If the curvature is approximately some positive constant (averaged over large areas), the Earth is a sphere. Guess the flat earthers were lying to us (unless NASA hacked your drone).

*If the curvature is approximately some negative constant (averaged over large areas), the Earth is a hyperbolic plane. Guess NASA and the flat earthers were part of a conspiracy to misled us.

*Otherwise, the Earth will be some other weird shape. I leave that for you to figure out.


The great thing about this method is that it is intrinsic. These are literally the definitions of flat and curved, so it leaves very little room to disagree with.
Also, if you only want to attempt to falsify some map, you just need to find some point of the world in which it is wrong. This has been successfully done for the maps proposed by the flat earthers, but no one has as yet found an incorrect distance or angle in the globe. Note though that this can never verify a map unless done over the entire Earth. If you find a contradiction, its wrong; if you don't, maybe you haven't looked hard enough.
A: The Earth being round is perhaps one of the simplest examples of a "hard" problem to prove convincingly. It requires the sum of evidence. We observe lots and lots of different phenomena, and individually none of them can trivially prove that Earth is round - as you can probably see from the rest of the answers here. But together they eliminate all alternatives, yet are all consistent with the hypothesis that the Earth is round.
There are many other problems that share this property, and the roundness of Earth is my favourite example of one. It really is harder than people give it credit for. It's "obvious" but it really isn't. The only truly obvious proof is to see it from space. All other evidence is at least somewhat circumstantial, it just happens to add up into a very consistent theory that explains everything we observe.
Importantly, the evidence we have is consistent not just conceptually, but also numerically. We can perform local experiments and all of them will arrive at the same, consistent and predictable numeric results.
A: One of more lesser known way of describing the earth's shape is by looking at night sky. In northern hemisphere, countries, they stars in night sky seem to move in clockwise direction wrt pole star. The more north you go, the more visible will be the effect.
In southern hemisphere countries, it is opposite. The stars seem to move in counterclockwise direction.
In equatorial region, the effect is very minutley visible, the stars seem to go east to west.
If you take a big ball like say basketball, and paste a small camera and you rotate ball in particular way, you will encounter similar results.
This was one of the ways, the explorers in past concluded the shape of earth, which is sphere.
Although we now know it's not perfectly spherical, but oblate spheriod, which means squashed at poles, and bulging at equator. This is due to centrifugal effect of rotation of earth.
Hope it helps
