introductory book on spectral sequences I have studied some basic homological algebra. But I can't send to get started on spectral sequences. I find Weibel and McCleary hard to understand.
Are there books or web resources that serve as good first introductions to spectral sequences? Thank you in advance!
 A: 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 ...
A: Tom Weston has a popular set of notes on spectral sequences on his website:
http://www.math.umass.edu/~weston/ep.html
A: I quite liked Gelfand-Manin's Methods of Homological Algebra.
For a tranlsation and expansion of the section in EGA you would definitely like Daniel Murfet's notes.
http://therisingsea.org/notes/SpectralSequences.pdf.
A: Chapters 9 and 10 of James F. Davis, Paul Kirk, Lecture Notes in Algebraic Topology, GSM #35, AMS shows how to apply spectral sequences to get many useful results 
rather than
focusing on their construction and proofs of their properties.
A: If you like David Mumford's writing style, i.e. clear, succinct and authoritative, try his notes in AG2:
http://www.math.upenn.edu/~chai/624_08/mumford-oda_chap7-8.pdf
A: Ken Brown's book, "Cohomology of groups" also gives a fairly readable introduction to spectral sequences.  The algebra is kept fairly simple here, and most of the discussion is about computing the homology of a double complex, and constructing the Lyndon-Hochschild-Serre spectral sequence.  So it doesn't go particularly deep, but it's a non-frightening place to start.
A: 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.
A: There's always McCleary's "A User's Guide to Spectral Sequences".
A: I'm going to have to agree with everyone who recommends Bott & Tu. That provided me with a good understanding of the basic setup. After I was comfortable with that, I moved on to Hilton & Stammbach's book "A Course in Homological Algebra" that did a good job of showing how the general idea works for Abelian categories. 
A: 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.
A: I would recommend that everyone's very first (zeroth?) introduction would be Timothy Chow's excellent short article You Could Have Invented Spectral Sequences. It doesn't give a lot of technical details, but it will definitely remove your fear before you start on a more advanced exposition.
Link: http://www.ams.org/notices/200601/fea-chow.pdf 
A: Also, Ravi Vakil has some nice notes about spectral sequences on his website: math.stanford.edu/~vakil/0708-216/216ss.pdf
[EDIT by DG: These notes have meanwhile become (in an updated version) a section (currently 1.7, as of 29 December 2015) of Ravi Vakil's The Rising Sea: Foundations Of Algebraic Geometry Notes.]
A: There's also a rather nice chapter in "The Heart of Cohomology" by Goro Kato. It could however be argued that I'm biased since I'm the one who typeset the book :)
Another good starting point is Timothy Chow's "http://www.ams.org/notices/200601/fea-chow.pdf". 
A: The book which cured my fear of spectral sequences is "Cohomology Operations and Applications in Homotopy Theory" by Mosher and Tangora. It only touches applications in topology, and by todays standards it would be considered very basic; the upside of this is that a lot of the material is passed in the exercises (another upside is that it's $10 on Amazon).
A: I am appending a list of references I found useful.
MR0243527 (39 #4848)  Mitchell, Barry . Spectral sequences for the layman. Amer. Math. Monthly  76  1969 599--605.
MR1721118 (2000m:55003)  McCleary, John . A history of spectral sequences: origins to 1953. History of topology, QA611.A3 H57 1999  631--663, North-Holland, Amsterdam,  1999.
Matthew Greenberg:Spectral sequences http://www.math.mcgill.ca/goren/SeminarOnCohomology/specseq.pdf
Tom Weston:Inflation-Restriction sequence http://www.math.mcgill.ca/goren/SeminarOnCohomology/infres.pdf
http://www.cmi.ac.in/~suman/academic/pranab/node2.html
Romero, Rubio, Sergeraert : Computing Spectral Sequences : http://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Ana-JSC.pdf
David Brown Serre spectral sequences and applications https://www.math.emory.edu/~dzb/math/papers/sss/
A: My Knowledge of spectral sequences is coming from the chapter 2 of book "Algebraic geometry" by Fu Lei. I think it is detailed and very easy to read. It is not theoretical oriented by organized for application of spectral sequences.
A: The more nice I remember was also my first meet by HA: Rotman "Introduction to homological ALgebra"
A: This is probably irresponsible but I decided to place a link to a raw pdf file entitled "Self-dual formulation of homology, exact couples and spectral sequences". It is a beamer presentation and it is awful: it does not contain a single word, just a pretentious panoply of diagrams.
The reason I decided to do this: I think the primary drawback of spectral sequences is the nightmare of indices, which is moreover not justified at all. The essential content of spectral sequences does not have anything to do with gradings a priori. So I decided to work out the construction of exact couples in the non-graded setup.
The result was spectacular in one respect: it shows that even if you get rid of gradings, the thing is still quite complicated :D
Still, I find it amusing to see what remains when you throw all the grading stuff out.
A: There are Uwe Jannsen's lecture notes on étale cohomology: http://www.mathematik.uni-regensburg.de/Jannsen/Etale-gesamt-eng.pdf Spectral sequences are covered in section 6.
