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I urge you to read Hatcher's book (or better, tom Dieck's book published by EMS) immediately. I can't even imagine learning homological algebra without a bunch of concrete topological examples to compute with. That sounds confusing.

Homological algebra without algebraic topology or geometry sounds like the driest possible exercise in pointlessness.

If you already understand homological algebra, Peter May’s AT book might get you where you want to be quickly. It’s free on his website.

good advice