Learning Homology and Cohomology

You can't go wrong reading the math bibliography at the Chicago undergraduate mathematics page. Personally I read though Rotman's book and found it suited me; but I tend to think less geometrically than is perhaps ideal. The more geometric reader would probably prefer Hatcher's book.

I should probably mention Tammo Tom Dieck's new book Algebraic Topology.

I'll just quote Hatcher himself about this book

Its viewpoint is fairly homotopy-theoretic, as in May's book, and it has a similar density coefficient that some commenters here seem to like. What really impressed me about the book is that in the last few chapters the author manages to give the first ever non-spectral-sequence proofs of some deep and fundamental theorems like Serre's theorem that the homotopy groups of spheres are finitely generated, and Serre's calculation of all the non-torsion. Another is the Hirzebruch signature theorem, the very last theorem in the book. These results are 50 years old, yet apparently no one had previously seen how to prove them without spectral sequences. Of course, spectral sequences are important things that serious topologists should know about, and their use cannot always be avoided, but it's illuminating to see when they are needed and when they are not. Whenever I get around to a second edition of my book I'll have to include tom Dieck's new approach, and I think one can go even further and develop the basic framework of rational homotopy theory without spectral sequences.

There is a review from MathSciNet located here. Summary is - a book at a high level, but very thorough and useful.