As I understand it, a good bit of the problem lies in the assumptions concerning space and time. General relativity uses Riemannian geometry as its basis, and that is a smooth space and time. Quantum mechanics, on the other hand, once you "zoom in" enough, assumes that space and time are choppy and frothy. String theory is one theory among many that attempts to reconcile the two. It has actually come upon hard times - string theorists are having a hard time getting jobs, as I understand it. String theory has no experimental basis, and that's its problem.
Relevant courses: quantum mechanics, general relativity and cosmology, quantum field theory, all at the graduate level.