ct.category theory - A good place where to learn about derived functors

I have to agree strongly with Ame's answer, in part. Weibel is a great place to go for the formalism. Once you have a little bit of the formalism, though, where to go depends on interests. To really see a lot of the power of derived functors, Hartshorne chapter 3 has some good theorems and exercises using them (though he does bounce back and forth with Cech cohomology...) and Huybrecht's "Fourier Mukai Transforms in Algebraic Geometry" book is very clear (at least to me) in the first few chapters, where he discusses derived categories (Note: I do disagree with Harry, this is a SECOND step, not a first for most) and makes extensive use of them, as he's interested in talking about derived categories, and they're the most natural thing in the universe there.

Also, the following papers might help:

Fourier Mukai Transforms and Applications to String Theory talks about how FM transforms (which are compositions of derived functors) help in string theory

Derived categories for the working mathematician This is a wonderful paper, very clear about what the derived category is, and might help in conjunction with the books above.