Reference for spectral sequences

I like the exposition in the appendix to Eisenbud's Commutative Algebra, which does everything in a kind of friendly way via exact couples. It doesn't do the Grothendieck spectral sequence, though, but that's in many other places (e.g. Lang's Algebra), and essentially follows (after some homological stuff) from the spectral sequence of a double complex. (I should confess that I've not myself needed to know too much about spectral sequences at the moment (beyond the Leray spectral sequence and the Cech-to-derived functor one in sheaf cohomology), and my knowledge of them is correspondingly limited.)

EGA III has several applications of spectral sequences in algebraic geometry. The very first result, that the higher quasi-coherent cohomology vanishes on an affine scheme, implicitly uses the Cech-to-derived functor spectral sequence since the computation given is that of the Cech cohomology (but he doesn't actually go through the details--I think they're in Godement's Theorie de faisceaux, but the deduction of it as a special case of the Grothendieck's spectral sequence is in Milne's online notes on etale cohomology). Another application in EGA III is the proof of the proper mapping theorem: that a proper map of noetherian schemes preserves coherence. The argument is very pretty, and starts in the standard case by doing this for a projective morphism (which is essentially the calculation, as in Hartshorne, of the cohomology of the standard line bundles on $\mathbb{P}^n$). But to deduce it for a proper morphism, the real fun starts when you "approximate" a proper $f: X \to Y$ morphism by a projective one (using Chow's lemma) and then show that there is a "significant" sheaf on $X$ which pushes forward to $Y$ coherently using the projective morphism and the Grothendieck spectral sequence between various push-forwards. (There's also a devissage argument that I'm skipping here.)

I really like the way Bott & Tu develop spectral sequences in "Differential Forms in Algebraic Topology". It would be an especially good choice if you happen to have some familiarity with de Rham cohomology; they begin by just sort of easing you into the whole idea by proving the isomorphism of de Rham cohomology and Cech cohomology by constructing a double complex whose cohomology is isomorphic to both of those.

Of course, they only give uses for spectral sequences in topology. I think they mainly introduce them to be able to give the spectral sequences for the homology and cohomology of the total space of a fibration. In this application there is particularly nice geometric intuition for the algebraic machinery, which I found very useful for trying to get to the bottom of what these things really do.

I really like the last chapter of Aluffi's book "Algebra: Chapter 0."

Perhaps one of the canonical references (especially to see tons of applications) is John McCleary's "A User's Guide to Spectral Sequences."