introductory book on spectral sequences

Many of the references that people have mentioned are very nice, but the brutal truth is that you have to work very hard through some basic examples before it really makes sense.

Take a complex $K=K^\bullet$ with a two step filtration $F^1\subset F^0=K$, the spectral sequence contains no more information than is contained in the long exact sequence associated to $$0 \to F^1\to F^0\to (F^1/F^0)\to 0$$ Now consider a three step filtration $F^2\subset F^1\subset F^0=K$, write down all the short exact sequences you can and see what you get. The game is to somehow relate $H^*(K)$ to $H^*(F^i/F^{i+1})$. Suppose you know these are zero, is $H^*(K)=0$? Once you've mastered that then ...

Bott and Tu, "Differential forms in Algebraic Topology" has some very nice exposition on spectral sequences. It has a fairly geometrical starting point, motivating the whole subject by generalizing the Meyer-Vietoris sequence to more complicated coverings and relating Cech cohomology to de Rham cohomology.

I found Allen Hatcher's notes (which can be found here) clear and very helpful. They're not complete, but what is there is excellent, I think.