I am interested in studying Atiyah Singer Index Theorem and Spin Geometry and would like to study Clifford Algebras and their representations for this purpose. I have a book 'Clifford Algebras : An introduction' by D.J.H. Garling. However I picked this book randomly and do not know if it is a good book. What are other standard references for this subject ? Would the material covered in Garling's book be sufficient algebraic background for study of Index Theorem and Spin Geometry ? I have some knowledge of graduate level algebra (Groups, Rings, Fields, Galois Theory, Commutative Algebra). Is it a good idea to study representation theory of Lie Groups and Lie Algebras say from Brian Hall's book before studying Clifford Algebras ?
As mentioned above, Spin Geometry by Lawson and Michelson is a good reference. For those so inclined, I would also recommend the second edition of Geometry, Topology and Physics by Miko Nakahara, which devotes the last chapter to a proof of the Atiyah-Singer index theorem based on supersymmetric quantum mechanics.