I would like to study Hatcher's book, Algebraic Topology - in particular the fundamental group and introductory homotopy theory. I haven't had formal instruction in algebra or topology (my background is primarily in analysis). I've read through the first five chapters of Munkres' Topology and have a fairly good grasp on everything except the proof of Tychonoff's theorem - is that sufficient, or should I continue reading Munkres?
As for algebra, my knowledge is considerably less; it is mostly what I have taught myself, but I've never seriously studied it. I'm familiar with basic notions of group theory but not so much with the major theorems. I am assuming this is where I should focus my efforts on in preparing to study Hatcher's book. What are some topics that I should be familiar with, and some texts to study those from? Would Dummit and Foote be a suitable choice, or should I seek something not quite so heavy? I would like to say I am "mathematically mature," just not specifically familiar with algebra.
I should stress that I'm not looking to become an expert in algebraic topology, just enough to study the fundamental concepts and theorems. My question is mostly whether I should focus more on algebra or topology in my preparation.