The book everyone seems to use to study Algebraic Geometry is Hartshorne's book. However, I hear a good number of people saying that this book totally misses the functorial point of view. Hence, could you please recommend a good source to learn AG using the Functor of Points approach? Thanks!!
Let me second the recommendation of Mumford's Lectures on curves on an algebraic surface, which is really fantastic. Mumford's Red Book, although at a more basic level, is also very good.
In a slighty different direction: if you have succeeded in solving a large number of Hartshorne questions, then why don't you just begin reading some research papers? If you want to learn the functor-of-points view-point, choose papers which emphasize this. Ultimately, I think that this will be more productive than looking for comprehensive and self-contained texts.
Try out "Groupes algébriques" (1970) by Michel Demazure and Pierre Gabriel. In the beginnung so called $Z$-functors (which are just functors from Rings to Sets, under appropriate set-theoretic assumptions) are studied "geometrically". In particular, you can define a quasi-coherent module on it, etc.
Dear Brian, it seems that algebraic geometers who adopt your favoured approach are essentially specialists in algebraic groups.
My favourite example would be Jantzen's Representations of Algebraic groups", Academic Press 1987, in which all of Chapter 1 (18 pages) is devoted to the functor approach you require. Let me emphasize that Jantzen doesn't limit himself to affine schemes nor to group schemes. He considers completely general schemes defined as local functors admitting an open covering (in the functor sense!) consisting of affine schemes . I am sure you'll love the ingenious but natural definitions of open subfunctor, closed subfunctor, base change ... introduced in this meaty chapter: good luck!