Is the chapter on groups in Herstein's Topics in Algebra enough to get started? What bare minimum Algebra should I be comfortable with, before studying Algebraic Topology?
Probably not, when covering things like the characterization of the fundamental groups of compact surfaces a couple of facts about free groups are used, also when calculating the abelianizations of the aforementioned groups.
However, I recommend that you learn these as they appear while you are reading Hatcher. I think that Hatcher's book is going to require a lot of work regardless of whether you know a lot of group theory or not. But it is a great book, and I think that learning the group theory stuff as you advance through the book is a good method.
As to what the "bare minimum" is: I thing that chapter on Herstein is more than enough.