Of course the canonical (graduate-level?) algebraic topology text is Hatcher's, and while I've found it pretty difficult to jump into as a beginner, if you've somehow overlooked it it's probably worth checking out. (It's available for free online, I believe.)
This archived sci.math.thread (now here, hat tip to user pjfmc) helped me out with understanding what "holes" are, really. I seem to recall that John Baez has written other stuff on homology I've found very intuitive and useful, but I don't feel like digging through the TWF archives right now to find it -- I may edit this later.
Finally, if you have access to a copy of the PCM, Burt Totaro's article on algebraic topology is quite well-written and intuitive, although there's probably more material on homotopy than (co)homology there. His references include Hatcher as well as Armstrong's Basic Topology at a lower level (which I own, and while I quite like it in general, I'm not a fan of the chapter on homology), Bott and Tu's Differential Forms in Algebraic Topology and Milnor's stuff.
