I don't know anything about group cohomology and I'd like to.
What is the best text to learn this subject?
I'd prefer as soft an introduction as possible - that is, lots of motivation, lots of examples, slow moving development of intuition. (Something like Simmons' Intro to Category Theory, if you've read that.) I have a background in finite group theory and some character theory. The text should actually show how to compute group cohomologies, preferably not too far in, something I have not found in the group cohomology books in my University library.
A textbook would be great, but good lecture notes or papers would be fine too.
For many reasons, I would suggest Weibel's "Intro to Homological Algebra", because it puts things like "group cohomology" into a somewhat larger context, enabling comparisons to other things... Perhaps no reason to be a complete slave to the ordering of topics therein... but to see that "group cohomology" consists of the right-derived functors of the "fixed-vector" functor (and "group homology" of left-derived functors of "co-fixed-vector" functor), as an example comparable to Lie-algebra (co-) homology, and many others, makes it easier to understand each particular example.