Assume you're talking to someone ignorant of the basic facts of astronomy.
How would you prove to them that Earth orbits the Sun? Similarly, how would you prove to them that the Moon orbits Earth?
I originally had something about the constellations changing in the sky to show that the Earth orbits the sun, but that would still be the case if the Sun orbited the Earth instead. Now that I think about it, there is one thing that conclusively proves that the Earth orbits the sun: parallax. Over the course of one year many of the stars will move relative to each other. At the end of the year they will be back where they started. This is because the Earth moves around in a 2AU diameter circle, so that six months from your first observation, you'll be standing 2AU away from where you were then, and are viewing the stars from a (slightly, but observably) different angle.
To show that the moon orbits the earth you could observe its location at the same time every night, and see that it moves, and is always nearly the same distance from earth. It never goes into a retrograde motion. Assuming the earth is spherical, the only way this could be true is if the moon orbits the earth. You might also take the phases of the moon into account and model the Sun-Earth-Moon system to explain it.
So called 'stellar aberration',the shifting of the apparent positions of stars by up to 20 arc seconds towards the direction the Earth is going in its orbit, was the first method available to 'simple'equipment in the 19th century, namely transit telescopes.
By the early 20th century, the annual variations of the Doppler shift of stellar spectral lines, caused by the 30 km/sec Earth's orbital motion, was readily detectable.
Now, I would say the easiest way would be observations of the annual Doppler shift of the 21-cm line of galactic neutral atomic hydrogen, by amateur radio astronomers. Perhaps others can now measure annual variations of the apparent temperature of the cosmic microwave background.
Stellar parallax, while a very small effect (less than one arc second) is easy to measure with modern amateur equipment, by imaging the same piece of sky with a nearby star three times, with six months between images.
Observationally, using resources available to students, it's actually pretty hard to dissuade a determined advocate of, say, a Tychonic system.
One approach is to show how well Newton's laws describe the motions of the planets. This requires a lot of time, both for data collection and for teaching Newton's laws.
The phases of the planets, especially venus, make it easy to work out the 3-dimensional positions of the Sun, Earth, and other planets.
Keeping in mind Eric's answer and my comment about it (basically, they are in the most generous sense mathematically equivalent, and it's only a matter of taste which is used), the only way to "prove" which is the superior point of view is via Occam's Razor arguments.
I think any layman's grasp of everyday physics, combined with the relative sizes/masses of the Sun, Earth, and Moon, make it clear that the smaller objects being tossed around by the relatively stationary, larger objects makes the most sense, and indeed it is much easier to describe mathematically.
Depending on your view of the question, you could either pull out a textbook and quote the quantities in question as givens, or carry out experiments accessible even to the ancients, in many cases. http://en.wikipedia.org/wiki/Astronomical_unit
If you're lucky enough to get someone who has seen the Newton film-strip/video/whatever-it-was showing cannons firing from towers and where they fall to earth, and when they don't, you can use that as a foundation. In my experience, most people have had that at some point in their schooling even if they never took the concepts farther.
If they haven't had that, sketching it out should work. The early concepts are intuitively obvious. If they take that on faith (bad scientist! No biscuit! But then, these are laymen) then the later concepts are easier to work in. Great! Now we have a concept of 'orbit'.
If you're very lucky, you might be able to jump from here to the following:
The above makes a big assumption for the sake of simplification: that when I say "the Earth orbits the Sun", that the barycenter is inside the Sun somewhere, and inside the Earth for "the Sun orbits the Earth". It also heavily implies that lighter bodies are in orbit around heavier bodies, and by locating the more massive body you also locate the body being orbited. Again, a great simplification but one likely to be accepted on faith by the layman.
Each of the statements in the above may be drilled into by the curious. The mass of the Earth? Geology has given us a good idea of the elemental concentration of the Earth and from there we can estimate mass. Orbital period? It's a year. Distance to the Sun? Parallax methods during solar eclipses, which also would give us the diameter of the sun.
If they don't go down that easy, time to educate them in more complex orbital dynamics.
It's time to work in the concept of barycenter. Point out that the orbit is actually around the center of mass of the dual system. Draw a line between the center of mass of both objects. The center of mass of the system is the point on that line where both sides of the line would be equally balanced if placed on a fulcrum. It is actually about this point that objects orbit. Now spin the line on that fulcrum point. The movement of both ends describe the motion of both bodies, the smaller body (such as the moon) appears to move in a circle around the larger body.
At this point a diversion into how the Moon influences tides and how that relates to barycenter could be a good thing. Depends on the audience.
Great! Now we have the concept that two bodies in an orbital relationship both move, as well as the concept of barycenter. Now to figure out if the Earth is the one that only moves a little.
For Earth and a man-made satellite this is nigh-indistinguishable from the center of the Earth. For the Earth and the Moon the barycenter is most definitely not at the center of the Earth, but not outside of it. For Pluto/Charon, it IS outside. The mass of both entities plays a key part in this dance.
For Earth/Sun we have a good idea of the mass of the Earth thanks to geology and geochemistry. We also have the orbital period. And we also have the distance between the Earth and the Sun. Given these numbers there are two values for the mass of the sun that can solve the equations; one where the barycenter is closer to Earth and another one where it is closer to the Sun. To find out which is which we must get more clues as to the mass of the Sun. Thanks to parallax and spectroscopy we know the diameter of the Sun and its chemical makeup, which make it pretty clear that the Sun is a lot more massive than the Earth. Therefore, the barycenter is closer to the Sun, and the Earth orbits the Sun.
The ultimate concept you're working towards is, "When two objects of such greatly disparate mass as the Sun and the Earth are in an orbital relationship, the barycenter of their orbits will be within the mass of the larger object. From the outside it looks as if the smaller object is in orbit around the larger object".
If I understand the question right, we suppose we want to prove to someone that the earth orbits the sun. I'm not quite sure that' the case from a scientific point of view.
Literally speaking, we can choose any reference frame we like and thus prove a heliocentric system or a or a geocentric.
The struggle, so violent in the early days of science, between the views of Ptolemy and Copernicus would then be quite meaningless. Either CS [Coordinate System] could be used with equal justification. The two sentences, ‘the sun is at rest and the earth moves’ or ‘the sun moves and the earth is at rest,’ would simply mean two different conventions concerning two different CS. Albert Einstein
It's a only a mathematics problem? In the part of solving the motion equations it is.But from philosophical and physical meaning it's another subject.
Philosophical:I will only make a reference to Hubble ..."Such a condition would imply that we occupy a unique position in the universe, analogous, in a sense, to the ancient conception of a central Earth...This hypothesis cannot be disproved, but it is unwelcome and would only be accepted as a last resort in order to save the phenomena. Therefore we disregard this possibility.... the unwelcome position of a favored location must be avoided at all costs.... such a favored position is intolerable...Therefore, in order to restore homogeneity, and to escape the horror of a unique position...must be compensated by spatial curvature. There seems to be no other escape”(Hubble, The Observational Approach to Cosmology)
The famous astronomer Edwin Hubble published on 1937 a study on the cosmological model of the universe, under the title “The Observational Approach to Cosmology”. In the data published in that study it was evident that Earth apperared like having a “unique” position in the cosmos, i.e. that it was in the center or very close to it. However Hubble chose not to accept that unique position based on philosophical propositions (principles) that be believed in. In particular and even though the nebula distribution showed that Earth should be in a center position, he discarded that idea based on the “principle” that we are not unique (so it is illogical to say that we are in a priviledged center position in the Universe). In order to accomodate that “principle” he added some corrective factors to his equations! As simple as that! No hard data, no scientific analysis – a plain philosophical choice was the basis of the choice of heliocentricity over geocentricity!
"The departures from uniformity are positive; the numbers of nebulae increase faster than the volume of space through which they are scattered. Thus the density of the nebular distribution increases outwards, symmetrically in all directions, leaving the observer in a unique position. Such a favoured position, of course, is intolerable; moreover, it represents a discrepancy with the theory, because the theory postulates homogeneity. Therefore, in order to restore homogeneity, and to escape the horror of a unique position, the departures from uniformity, which are introduced by the recession factors, must be compensated by the second term representing effects of spatial curvature."
And it goes like this. Also, it's no secret that the ancient Greeks had, because of Aristotle, a geocentric point of view, and where able to predict solar eclipses.
George Ellis proposed a “semi-geocentric” model universe that contains a naked singularity as a recycling mechanism, which he claims gives almost as good a description of the real universe as the conventional model. In an article in Nature Ellis proposes that the universe is like a cylinder-shaped universe with two “centers”, with the Earth is located on one side and a naked singularity on the other. There is no cosmic inflation – the galaxies are arranged very unevenly, with a great deal of material crowded round the singularity, and very little near the Earth. The effect of such a distribution of matter is to produce a red shift of light that, at the Earth, has the same characteristics as if the galaxies were receding. In particular, the Universe seems to be expanding ever faster — a phenomenon generally ascribed to the influence of ‘dark energy’. But Ellis suggests that the observed acceleration be a trick of the light in an inhomogeneous Universe and if we accept his geocentric model, we could do away with dark energy by imagining that we lived at the center of a spherical inhomogeneity.
As I understand it, and I'm closing with this, we haven't proved the existence of a subjective point of reference in the universe(the systems we construct are always a product of convention - we say were the zero is). And, either we will change relativity or we shall continue with this basic understanding as mentioned. What we shall choose as a reference frame will be a function of a future discovery or of a choice.
So, we don't want to teach that the earth orbits the sun. We will teach first under what assumption the earth moves around the sun or we could just say that what you see(the sun moving around the earth) is totally also correct(again under a sum of assumptions).
Hope this helped.
PS You could do some search in the web about the subject or Ellis.