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.