A useful reference is W.S. Massey: "A basic course in algebraic topology". Anything left unclear by this book can probably be found in J.J. Rotman: "An introduction to algebraic topology", or vice versa... It's best to get an idea what homology is about from Seifert & Threlfall and then to learn the techniques of relative cohomology from more modern books.

By separating the machinery from the objects of interest algebraic topology can be made more powerful, see for instance C.A. Weibel: "An introduction to homological algebra" for an overview (more details in other books by J.J. Rotman,  L.R. Vermani, etc.).

For the basics of the related subject of homotopy see chapter 3 of "Lecture notes on elementary topology and geometry" by Singer and Thorpe

Here's a free downloadable introductory textbook:  A. Hatcher: Algebraic Topology