The best books have already been mentioned - these are the books by Weibel and by Gelfand-Manin.
A rough guideline to reading these book is as follows:
One picks up elementary facts (snake lemma, complexes, homology, abelian categories) from algebra/algebraic topology courses; then, on the intermediate level, one uses most of Weibel's book to learn yoga of derived functors and spectral sequences in a hands-on way (chapters 2-5, then 6-9 are various specific applications which you might read later on when you need it); finally, for the cleanest modern account of derived categories machinery one reads chapter 3 of Gelfand-Manin. (I personally found chapter 4 on triangulated categories very helpful in my work too).
As far as your specific needs go:
1) Some algebra textbooks have this material (my favorite being Aluffi "Algebra. Chapter 0"). Appendix and chapter 1 in Weibel and chapter 2 of Gelfand-Manin explain abelian categories in depth, albeit in a rather dry fashion. For triangulated categories read chapter 4 of Gelfand-Manin.
2) Both Weibel and Gelfand-Manin have some sketchy remarks on classic algebraic topology, but maybe it is better to directly use algebraic topology textbook for motivation, e.g. May's book "A concise course in algebraic topology". However, both textbooks contain nice accounts of simplicial approach to algebraic topology.
3) Chapter 2 of Gelfand-Manin and chapters 1,2 of Weibel.
4) Sheaf theory is scattered over Gelfand-Manin's book and despite being conceptually explained really well, it might be too hard for a beginner. It might be better to go directly to an introductory text in algebraic geometry such as Vakil's notes or Hartshorne. Group and Galois cohomology at some introductory level could be found in chapter 6 of Weibel.