I have been trying to learn about spin geometry, Dirac operators, and index theory by reading Lawson/Michelson's "Spin Geometry" and Friedrich's "Dirac Operators in Riemannian Geometry." Both are abstract, and basically no explicit examples are worked all the way through.

For example, I have been trying to find the spinor bundles, Dirac operators, and various indices for relatively simple manifolds: spheres and tori. However often these computations are detailed and even when I get to the end, it's not clear that I've done it correctly.

Is there another book, or perhaps online notes, which have a bunch of examples worked through in detail so that I can make sure what I'm doing is correct and also have a bank of examples to look at as I progress?