I'm still just a beginner in algebraic topology, but there's a specific problem I'd like to understand, which is how to classify maps from one space into another up to homotopy -- for instance, I've really enjoyed learning about the Pontryagin-Thom construction which yields homotopy classification of maps into S^2. For some applications that I'm interested in, it turns out that the homotopy classification of maps from manifolds into homogeneous spaces (of strictly lower dimension, if that helps) are of interest.
I guess what I'm asking is for a pointer in the right direction, since algebraic topology is such a large subject. I've read scattered results here and there on specific examples of the above, but I haven't found any systematic way of thinking about it yet. Is there one? Someone once said "equivariant cohomology" to me, is that useful?