I'll just mention some sources that are not already in the other question indicated by Franz's answer, in case you need to become more comfortable with the derived machinery itself. (The book by Huybrechts on Fourier-Mukai Transforms in the other answer will probably be the best, depending on your interests, and there is also "Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics", by Bartocci, Bruzzo, and Ruipérez that contains additional information and good references).

So here are the other references that might help with derived categories themselves:

I think most of the texts or papers treating derived categories applied to algebraic geometry are a bit too terse when it comes to derived categories themselves. There is a nice book called "Interactions Between Homotopy Theory and Algebra", which is from a summer school that was held in Chicago, and in this book there are two nice lectures by Henning Krause on derivated categories. His second lecture actually consists of about 40 exercises, which might help you with understanding the machinery of derived categories without having to worry about how they work in algebraic geometry yet. These two chapters are also available on the arXiv (Lecture, Exercises), though I recommend the whole book too.

Yet another book is the edited volume "Handbook of Tilting Theory" (LMS 332), which is a series of longer (most introductory papers) on tilting theory, so you'll see plenty of applications to module categories and representation theory, but there is also a chapter on the Fourier-Mukai transform there.

In the book "Derived Equivalences for Group Rings" (König, Zimmerman, et al.), there are several chapters that include introductions to aspects of derived categories including unbounded derived categories and many examples, which might also be useful.

thederived category $D(X)$ of a scheme or stack $X$, although this notation is used quite often. But it can refer to the bounded, bounded below, bounded above derived category of the abelian category of Zariski, étale, lisse-étale etc. quasi-coherent or coherent sheaves on $X$ ... – Martin Brandenburg Jun 17 '13 at 15:31