Let me begin by citing a site: here you will find, among other things, a free downloadable version of an algebraic topology book, offered by its author. It looks pretty good.
Someday I’d like to write an introduction to topology (a post! not a book!), but trying to do it now is taking me too far out of my comfort zone: I am reasonably familiar with general (also called point-set) topology, but I am rather ignorant of algebraic topology; I am reasonably familiar with differential geometry, but differential topology is a different and unknown beast. I have opinions about how topology hangs together, but when I try to be precise, I find that I’m not sure I can justify my opinions. I’d rather get it more right later than get it wrong now.
Remember that the path from ignorance to knowledge in any subject is not straight and true, but is almost always rather zigzagged. One seems to learn things by a method of successive approximations to the truth.
William S. Massey, Algebraic Topology: An Introduction. p xiii.