TITLE: An "advanced beginner's" book on algebraic topology?
QUESTION [45 upvotes]: It has so happened that I have come this far knowing nothing on the subject of algebraic topology (as in homology theories of topological spaces and their applications). I've decided to finally read up on that during the summer.
Seemingly, however, the authors of most books for beginners are hesitant to make use of nontrivial homological algebra and category theory, which, if I'm not mistaken, could be used to speed up and at the same time clarify the presentation. I, on the other hand, would dare say to be somewhat familiar with these disciplines. (I'm, to different degrees, acquainted with derived functors, spectral sequences, derived categories as well as sheaf cohomology and Lie algebra/group cohomology.)
Thus, what I'm looking for is an introduction to algebraic topology the author of which readily employs the above concepts when appropriate.

REPLY [5 votes]: You have many good suggestions already. Another book which you might enjoy is "Cohomology Operations and Applications in Homotopy Theory" by Mosher and Tangora.
This is not really a beginner's book per-se, as it assumes a basic knowledge of ordinary cohomology from the start. However it has a lot to recommend it, including brevity, affordability and concreteness (the focus is on applications of cohomology theory to calculations of the homotopy groups of spheres). It also seems to meet your criteria in that it gets quickly to the deeper applications of homological algebra and spectral sequences in homotopy theory.