TITLE: Books for studying Dirac Operators, Atiyah-Singer Index Theorem, Heat Kernels
QUESTION [15 upvotes]: I am interested in learning about Dirac operators, Heat Kernels and their role in Atiyah-Singer Index Theorem. From various sources (including this very helpful question), I have come to know of various references :
(i) Spin Geometry by Lawson & Michelsohn
(ii) Heat Kernels and Dirac Operators by Berline, Getzler and Vergne
(iii) Dirac operators and spectral geometry by Giampiero Esposito
(iv) Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem by Peter Gilkey
and (v) The laplacian on a Riemannian Manifold by Rosenberg, 
however I am having difficulty deciding which one or two to study between these. I prefer reading books that start from basics but eventually cover the core aspects of the subject at fairly advanced level. Please advise which one or two of the above should I study such that the intersection of the selected books is minimal and the union is maximal. Any comments about the above-listed books will be very helpful.
My background is : Analysis (As covered in 'Principles of Mathematical Analysis' by Walter Rudin but not much of measure theory), Algbera ( 1 year grad level course based on Serge Lang), Manifold Theory and Differential Geometry (Differential forms, de-Rham theory, Riemannian metrics, Geodesics, Connections, Curvature, Vector bundles & Characteristic classes, Principle bundles ) but very little functional analysis (definition and basic properties of Banach and Hilbert spaces, and the four famous theorems) and almost no Algebraic Topology beyond the definition of fundamental group. By the time I plan to start studying ASIT, I am expecting to have studied Representation theory of Lie groups and Lie Algebras and may be a little bit of Clifford Algebras as well.Please advise what other prerequisites do I need to study the books listed above.In particular, do I need to learn more Functional Analysis and Algebraic Topology for this purpose ? I have studied the prefaces of the books I have listed but unfortunately the information about necessary prerequisites is not mentioned in most cases.
   Any other good references are welcome.
Note: I had earlier asked a related question here

REPLY [3 votes]: I have studied the whole book of Rosenberg and I can asure you that it is a very good choice to start with, I have studied some parts of the books of Gilkey and Lawson-Michelson too and I can asure you they are quite advanced, it would be hard to begin with them. I guess the book of Lawson-Michelson is better for you because it covers the classical proof of the Atiyah-Singer index theorem, which depends more on geometry, while Gilkey covers de analytic proof of the theorem, which depends very much on analysis. Anyway, if you prefer to choose the book of Gilkey, be sure to get the second edition, since it is fairly better than the first one.
Remark: The book of Gilkey starts in the context of elliptic complexes, which is more general than the context of Dirac operators and Dirac complexes