I am a graduate student working in differential topology, and I would like to start studying the Atiyah–Singer index theorem. What books, lecture notes, or other references would be suitable for a beginner?
I have searched Google for recommendations, but I feel that people who have already worked through the theorem may be able to suggest a better roadmap and point out common mistakes for someone approaching it for the first time.
I am familiar with Riemannian and complex geometry, vector bundles, Chern–Weil theory, functional analysis, and elementary PDEs.