I find Weibel very readable - once the interest is there. It's a good reference, and not as opaque as, for instance, the MacLane + Hilton-Stammbach couplet can be at points.
The interest, however, is something I blame my alma mater for. Once upon a time, Jan-Erik Roos went to Paris and studied with Grothendieck. When he got back, he got a professorship at Stockholm University without having finished his PhD. He promptly made sure that nowadays (when he's an Emeritus stalking the halls) there is not a single algebraist at Stockholm University without some sort of intuition for homological algebra.
So, my MSc advisor, J