What books/articles do you recommend for learning spectral sequences? I am interested in their applications to algebraic topology, particularly to understand the homology of fibre bundles. I have a good backgroud on differential geometry and a reasonable background on modules and algebras.

$\begingroup$ The Serre spectral sequence is the starter for most people, as far as I know. I found Spanier to be a good reference, but likely there are better and more modern references now. $\endgroup$ – Ryan Budney Mar 4 at 4:10

4$\begingroup$ I think Hatcher has some good notes. $\endgroup$ – Anubhav Mukherjee Mar 4 at 4:15

1$\begingroup$ Mosher and Tangora is a good (but perhaps oldfashioned) way to get started with the Serre spectral sequence. $\endgroup$ – John Palmieri Mar 4 at 6:03

$\begingroup$ I really like homotopical topology by fomenko and fuchs $\endgroup$ – Thomas Rot Mar 4 at 7:05

2$\begingroup$ Relevant questions: mathoverflow.net/questions/45036/…, mathoverflow.net/questions/23297/…, mathoverflow.net/questions/22188/… $\endgroup$ – David White Mar 4 at 12:28
My favourite reference for understanding spectral sequences is
Boardman, J. Michael. "Conditionally convergent spectral sequences." Contemporary Mathematics 239 (1999): 4984 (pdf).
I don't think I really understood a spectral sequence before reading it. After that, examples examples examples. The Serre spectral sequence is a good place to start (and Serre's original paper not the worst place to learn it).
The article by Timothy Chow, "You Could Have Invented Spectral Sequences", Notices of the AMS 53 2006 pp. 1519 (pdf) is a start.
Given your background and interests, you could try John McCleary's A User's Guide to Spectral Sequences, (Cambridge, 2001) doi:10.1017/CBO9780511626289.
More at the following question: Spectral Sequences reference.

2$\begingroup$ Doug Ravenel "accidentally" put his personal copy of the pdf of McCleary's book on an internetfacing location on his computer... $\endgroup$ – theHigherGeometer Mar 4 at 6:31
I have learnt it from Mosher and Tangora. But, perhaps John McCleary’s book is a reference to be mentioned.