Assume you're talking to someone ignorant of the basic facts of astronomy.
How would you prove to them that the Earth orbits the Sun? Similarly, how would you prove to them that the Moon orbits the 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.
The phases of the planets, especially venus, make it easy to work out the 3-dimensional positions of the Sun, Earth, and other planets.
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.
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.
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".