I actually think that the Hilbert scheme should be mentioned (and, if possible, proved to exist and discussed) as early as possible. It serves as a good example of a moduli space, and it exists! Plus, the infinitesimal study of the Hilbert scheme allows some deformation theory to be discussed (at least, the deformations of projective schemes inside projective space) which also helps explain, algebro-geometrically, what the normal sheaf really controls. Add to this the fact that a lot of research relies on moduli spaces these days (In particular, I know that people care about Hilbert schemes of points, and, if some GIT for PGL can be covered, it'll let you actually construct $\mathcal{M}_g$, which finishes the classification of curves that's given in chapter 1 of Hartshorne, though this is a bit more.)
Because you'll be wanting things fundamentally scheme theoretic, the first part of Kollár's "Rational Curves on Algebraic Varieties" might be a good reference for this stuff.
