A down-to-earth introduction to the uses of derived categories

I won't claim to be fluent in the language of derived categories, but I understand it and can make myself understood. For most people, that's the right level of proficiency. Since you already have plenty of references, let me instead share few thoughts about the relation between the old and new languages. This is an imperfect analogy, but in the older differential geometry literature, everything was written in coordinates leading to messy formulas ("the debauch of indices"). By contrast, modern treatments are coordinate free which is better much of time, although not all of the time. I tend to think of spectral sequences as writing things in coordinates; derived categories are coordinate free. (This obviously a stretch. In hindsight, this seems to be my answer to the question Thinking and Explaining as well.)

Let me spell this out. Given left exact functors $F:A\to B$ and $G:B\to C$ between abelian categories, under the usual assumptions, we get the Grothendieck spectral sequence $$E_2^{pq} = R^pF (R^qG M) \Rightarrow R^{p+q}F\circ G M$$ By constrast, in the derived category world we see a composition law $$\mathbb{R} F\circ \mathbb{R} G\cong \mathbb{R}F\circ G$$ For 3 or more functors, the last formula generalizes in the obvious way. On the spectral sequence side, we get something too horrible to comtemplate. Well no, let me comtemplate it: $$E_2^{pqr\ldots} = R^pF (R^qG (R^rH\ldots))$$ $$d_2^{2,-1,0,\ldots}: E_2^{pqr\ldots} \to E_2^{p+2,q-1,r,\ldots}$$ $$d_2^{0,2,-1\ldots}\ldots$$ $$\ldots$$ Don't get me wrong, spectral sequences are still useful, but not here.

I would suggest "Fourier-Mukai transforms in algebraic geometry" by Daniel Huybrechts.

"Derived Categories of Sheaves: A Skimming," by my colleague Andrei Caldararu, might do the trick. It's the notes from his lectures at the Snowbird summer school.