Assuming you already know undergraduate algebra (and maybe a little basic homological algebra already), the book Methods of Homological Algebra by Gelfand and Manin is a good source for the kinds of things in the first chapter or so of Kashiwara-Schapira, i.e. derived categories, derived functors. I can't remember how elementary its sheaf theory is, but a little background in sheaf theory wouldn't hurt either. Swan's book is probably a kinder starting point for sheaf theory then Bredon.
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