I expressed my frustration with Hartshorne's book a bit here:
The point is that many definitions in algebraic geometry are basically obtained by taking definitions from topology or algebra, translating them into "purely category theoretic language" and then using that definition as a substitute in the category of schemes.
In particular I unravel the definition of a separated morphism:
"A seperated morphism of schemes is one where the image of the diagonal is closed."
If we just replace "schemes" with "topological spaces", then this property for spaces says (after a little definition chasing)
"Any two distinct points which are identified by the morphism can be separated by disjoint open sets in the domain"
Thus a space is Hausdorff as a topological space iff the unique map to the one point space is separated. Before I worked through this I had no real reason to believe that separated morphisms were a natural concept. Why don't people ever talk about the topological analogue?
Another point of much confusion for me was the definition of derived functor cohomology. Why should we care about injective resolutions? Anton gives a great answer here:
Anton's line of thought is also beautifully developed in Gunter Harder's book "Lectures on Algebraic Geometry 1". The quick and dirty version is that cohomology should have nice properties (ses gives rise to les, etc) and acyclic resolutions compute cohomology. Hey! Injective objects are always acyclic (this is reasonable because they make ses's split). Thus injective resolutions are a nice generic thing to use.