Moshe has already addressed many points. You might be interested in Folland's Quantum Field Theory: a tourist guide for mathematicians. He tries to do as much things as possible in a mathematically rigorous fashion, and points out those points where this cannot be done.

As for the mathematical background: some familiarity with partial differential equations and the theory of distributions will be convenient.

This concerns "conventional" quantum field theory. You might also be interested in topological quantum field theory which is much more of a mathematical nature.