I have no doubt that Hatcher's is a great text, but it is not for me. It is clearly written for someone with some prior knowledge of these topics, and I have none. For example, in the first few pages he defines deformation retractions, mapping cylinders, and homotopies. I gained very little insight into what these actually are from reading.
I am looking for an alternative text. To give an idea of my background, I am most comfortable with analysis, moderately comfortable with topology (first five chapters of Munkres's text), and mildly comfortable with algebra (basic group and ring theory). I am hoping for a text that not only defines the terms, but motivates them and helps the reader to understand them. I am most interested in homotopy groups and manifolds, out of the "main branches of algebraic topology" listed on Wikipedia.
I'm aware this question has been asked before, but a cursory search didn't really find any good consensus.
Best Answer
If you want a more rigorous book with geometric motivation I recommend John M. Lee`s topological manifolds where he does a lot of stuff on covering spaces homologies and cohomologies. As a supplement you can next go to his book on Smooth Manifold to get to the differential case. I especially like his very through and rigorous introduction of quotient spaces/topologies and so on which are used very heavily and which Hatcher explains mostly in a very pictorial and unsatisfying way.
Try Tammo Dieck's Algebraic Topology. It is very detailed and a good supplement to Hatcher.I got my first exposure to algebraic topology from that book.