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 animperfect analogy, but in the older differential geometry literature, everything was writtenin coordinates leading to messy formulas ("the debauch of indices"). By contrast, moderntreatments 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; derivedcategories are coordinate free. (This obviously a stretch. In hindsight, this seems to be my answer to the questionThinking and Explainingas 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 thespectral sequence side, we get something too horrible to comtemplate. Well no, let mecomtemplate 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.