If I really had to pick one (and I do!), I would have to pick the Pythagorean theorem. I have a few reasons - including memories of friends giving hour-long proofs of this theorem, memories of high school math, memories of a seven-page paper on Pytagoreas, etc., etc. I guess the main reason is that it's one of the more commonly used theorems, and it comes in handy a lot in geometry - which is where I think I'll end up teaching first.
