ca.analysis and odes - Atiyah-Singer index theorem

You need to understand pseudodifferential operators if you want to understand the original statement of the full Atiyah-Singer index theorem. However, in most applications to differential geometry, only the theorem for twisted Dirac operators is needed. (One of the main results of Atiyah and Singer is that the Bott periodicity theorem - or rather, its generalization to vector bundles, the Thom isomorphism theorem for K-theory - reduces the general case to that of twisted Dirac operators.)

If you want to learn the theory of pseudodifferential operators, I recommend the original papers of Kohn and Nirenberg and Hörmander. This theory is not needed to prove the Atiyah-Singer index theorem: you can get away with the existence of an asymptotic solution of the heat equation. To see this in action, see the paper of McKean and Singer.

One advantage of the heat-kernel approach is that it is well-adapted to study the generalizations of the theory, such as the theory of analytic torsion and the family index theorem.