For what it's worth, I learned most of what I know of derived categories from the last chapter of Weibel's book on homological algebra (and in particular, doing the exercises there). However, I also often looked at Hartshorne's Residues and Duality, Gelfand-Manin, and occasionally derived categories for the working mathematician.
My goal at the time was to understand better things like Grothendieck duality.