Homological algebra has been at the center of twentieth century mathematics. It has widespread applications to topology, algebra and geometry. In this project, we will cover the classical approach via derived functors, and then move to the more modern approach, by derived categories.
Note: The best exposition, of the beginning of this subject, is still the 1955 paper by Grothendieck, Sur quelques points d'algebre homologique. This paper, as you undoubtedly guessed, is in French. If the student is willing to read French, that would be a great plus. But it is not entirely unavoidable; there are English versions.