I am surprised that no one mentioned this so far; I am only imagining that everyone thought it so natural that it escaped their mind.
Most "standard courses" would be following Hartshorne's book, I assume. It is a great loss that this book does not mention the "functor of points" view at all. It would maybe take 10 or 15 minutes to state and prove the Yoneda's lemma, and a little more time to mention the functor of points and the advantage of this point of view for applications to arithmetic geometry(points with values in a certain ring, base change, etc.), and more importantly for moduli problems. One could also give a definition of a fine moduli space and coarse moduli space, and as examples just mention the the moduli space of curves with marked points(but without proofs, of course).