Paolo Aluffi's Algebra Chapter 0 develops abstract algebra using Category theory from the very beginning. The exposition is very clear and teaches upto and including the derived functor approach to cohomology. The category theory developed here should be more than enough to study sheaves and schemes eventually.
In an answer to your earlier question, Julien Puydt points to (and I second his suggestion) the excellent text by Ravi Vakil. The category theory developed in this text is really all you need; in fact, there is no more than what is needed for purposes of getting started with AG.