So far the most comprehensive book looks like McCleary’s book: A User’s Guide to Spectral Sequences. It is also suitable for those interested in the algebraic viewpoint. W.S. Massey wrote a very positive review to this book.
A User’s Guide to Spectral Sequences (Cambridge Studies in Advanced Mathematics)
Another book is Rotman’s An Introduction to Homological Algebra (Universitext). This book is from a homological algebra viewpoint. Rotman has a nice easy-going style, that made his books very popular to read.
The classic book may be MacLane’s Homology (Classics in Mathematics). This may be harder to read (though to be honest all books on spectral sequences are hard).