Michael Artin

This is the book I learned algebra from. It is much more concrete than Aluffi, and is a much more classical wander through the basics of the subject. The chapter on Galois theory is a nice inclusion; possibly skip the modules chapter, though.

This is a really fantastic introduction to algebra. The author asks a bit more of the reader than some other books at this level, but that's for the best. To paraphrase one of my professors, this is "the mathematician's introductory algebra textbook."