Every year or so I make an attempt to "really" learn the Atiyah-Singer index theorem. I always find that I give up because my analysis background is too weak -- most of the sources spend a lot of time discussing the topology and algebra, but very little time on the analysis. Question : is there a "fun" source for reading about the appropriate parts of analysis?
I found Booss, Bleecker: "Topology and analysis, the Atiyah-Singer index formula and gauge-theoretic physics" (review) very beautifull and had read it just for fun. It is a very nice piece of exposition, motivates everything and demands from the reader only very little preknowledge.
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.
I know it may seem rather "old", but the notes from the IAS "Seminar on the Atiyah-Singer Index Theorem from back in 1965 (published by Princeton Univ. Press) may be just what you are looking for, since it covers all the analytic machinery in great detail. It was written to be easily accessible to a math graduate student who had a basic analysis course.
I first learnt about the Atiyah-Singer index theorem from Shanahan's Springer notes (638). I liked it because while developing the main theory, it went through the standard examples (Dirac, Dolbeaut, de Rham, signature) in some detail. At the time I was primarily interested in using index theory so wasn't so bothered about the details of the proof, but it does at least sketch the proof (it may do more, I don't remember and don't have a copy on my shelves) but I do remember that the words "pseudo-differential operator" occur which suggests that even if sketchy, the main points are all there. He also covers the equivariant theory.
Another place where it's put in context is Spin Geometry by Lawson and Michelsohn. That's quite nice because the whole theory of Dirac operators and Clifford algebras is developed from scratch so there are lots of "entry points" depending on whether or not you're more of an algebraist or geometer or other.
- The Atiyah-Singer Index theorem: MR487910
- Spin Geometry: MR1031992
There is also a "physicist's" proof of the index theorem. (No, really, keep reading!) Attached to every supersymmetric quantum field theory (or even quantum mechanics) there is something called the Witten index. There are quantum mechanical systems for which the Witten index coincides with the index of an elliptic operator (made out of the supercharges of the theory). It is not difficult to argue that the Witten index has a homotopy invariance property which allows one to compute it in different "temperature" regimes. At infinite temperature it simply counts the difference between the dimensions of the kernel and cokernel of the elliptic operator, whereas at zero temperature it can be evaluated to give an integral formula for the index in terms of characteristic classes. This can be made rigourous and the details can be found this paper of Getzler's.
Perhaps you will also like
"Heat kernels and Dirac operators" by Nicole Berline,Ezra Getzler,Michèle Vergne
(see the google books link here)
Perhaps not the "full" Atiyah-Singer index theorem, but that for Dirac-Operators. So if you are more a differential-geometer than an "analysis-guy", I think this is an appropriate reference.
(EDIT: As José Figuera-O'Farril remarked, AS for Dirac Operators should be enough to get AS for elliptic operators)
In addition another book by Gilkey (compare the post of José Figueroa-O'Farrill and the comment by Ryan Budney) should be very readable:
Invariance theory, the heat equation, and the Atiyah-Singer index theorem by Peter B. Gilkey
I'm reading the original paper right now and finding it a little terse in some places, but a good guide. "Spin Geometry" by Michelson and Lawson has a chapter that covers in great detail all the analysis you need for index theory as well as complete proofs of the index theorem in all of its forms. It also has some decent chapters on K-theory (What you really need is to understand the Thom Isomorphism in K-theory. Segal's "Equivariant K-Theory" also has a nice description.)
Hope this helps.
If you have access to Atiyah's collected works at your library, try taking a look at those. There are a few transcribed lectures and short expository papers where he explains the context and motivation of the theorems. (If I remember correctly, they are classified as miscellanea and appear in the first volume, but there might also be some in the second volume. I don't have it at hand to check.)
He writes beautifully, and for myself, I didn't feel like I "got" the index theorem until I read these.
EDIT: Oops, I didn't read your question carefully enough: you are mostly looking for the analysis part. In that case I would just second the recommendation of the relevant chapter in Warner or Wells.
If you're still interested in improving your background for understanding the ASIT, you can try with these lecture notes, from a course given at the Utrecht University by E.P van den Ban and M. Crainic.
The course spent time in the analytic part of (this) proof, going through pseudo-differential operators and it's symbols, and showing the role of Fourier Analysis in it's construction.