Victor Protsak suggests this, and I'll endorse it: the careful construction of the Grassmannian. This is a good example for 3 reasons. (1) It is extremely important. (2) It is a situation where it is both natural to work in local coordinates and with a global projective embedding, so students can practice transforming between the two perspectives. (3) It is small enough to do in full detail, but it usually isn't.
Ideally, this would include proving that the Grassmannian represents the functor of flat families of subspaces of a vector space. (Or quotient spaces, whichever you prefer.)