An "advanced beginner's" book on algebraic topology? 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.
 A: A recent book is tom Dieck's "Algebraic Topology", which is precisely written and quite comprehensive. But I've only skimmed it, so I'd be interested in more expert opinions.
A: I think you're describing Spanier.
Everyone I know who has seriously studied from Spanier swears by it- it's an absolute classic. The approach is exactly as you describe- algebraic topology for grown-ups. The treatment of homological algebra in it is extremely nice, and quite sophisticated.
A second, quite brilliant book along the same lines is Rotman. It's more geometric than Spanier, for those who like such things, and find it easier to read (although that's a matter of taste of course). Again, the treatment is unembarrassed to employ nontrivial homological algebra and category theory, in a good way. 
A: My suggestion: 
Algebraic Topology: Homotopy and Homology - by Robert M. Switzer
A: You might find Jeff Strom's new book attractive. Here is a review.
A: 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. 
A: It is somewhat jarring to hear of people who "know nothing about the homology theories of topological spaces and their applications" but are "familiar with homological algebra, category theory, spectral sequences (!!)" and the like. Certainly, this is a historically backwards position to be in, since a lot of these highly abstract theories with which you are familiar were almost entirely motivated by the concrete theory which you now wish to master.
Based on what you have said about your background, you will find Peter May's book "A Concise Course in Algebraic Topology" an appropriate read. Peter does not shy away from using categorical or homological machinery when dealing with this material, but also encourages his reader to become adept at the sort of calculations which yield insight into the nature of the subject. Amazingly, you can get the book freely off his website.
Also, to really hammer in the fact that you are engaging with a living, breathing, highly applicable subject, check out Kaczynski, Mischaikow and Mrozek's "Computational Homology" and Edelsbrunner and Harer's "Computational Topology" to find a breadth of applications of homology to physical and life sciences.
Update: The OP and others in a similar position may also be interested in my own upcoming book. You can find the cover here.
A: There's a great book called Lecture Notes in Algebraic Topology by Davis and Kirk which I highly recommend for advanced beginners, especially those who like the categorical viewpoint and homological algebra. I think the treatment in Spanier is a bit outdated. Davis and Kirk is written with an eye towards what you might learn next,  e.g. model categories. By the way, there's a pdf of it available for free here.
A: My sense is you haven't read Allen Hatcher's book closely enough.  I certainly need to go through it.
Jacob Lurie had nice Geometric Topology course a few years back, if you like that style.  Here's an intriguiging sounding course on Chromatic Homotopy Theory
Akhil Matthew took notes on a course by Michael Hopkins.  

You may wish to delve into the literature more directly.  Have you looked into Twisted K-Theory ?
