Books for studying Dirac Operators, Atiyah-Singer Index Theorem, Heat Kernels

I think the best introduction is "Dirac Operators: Yesterday and Today", the proceedings of a summer school edited by Bourguignon, Branson, Chamseddine, Hijazi and Stanton. It does not cover heat kernels, but gives a good understanding of the other topics - and does start from basics; there is a (short) chapter on differentiable manifolds. Also I think a good understanding of the contents here is prerequisite to getting much out of, say, the Lawson-Michelsohn book.

Beyond the prerequisites you mentioned, I think the only thing necessary is a certain understanding of cohomology groups. To start learning about heat kernels, timur is probably right in saying you should first learn some PDE.

In have worked for a longer time with the mentioned books of S.Rosenberg (v), P.Gilkey (iv) and Berline, Getzler and Vergne (ii). Since my interests were more related to heat kernels than Dirac Operators I want to comment from this point of view.

The book of S.Rosenberg is excellent to start with. It deals with the basics though trying to touch topics are quite complicated, so it is very good to start diving in.

(ii) is a great source, too. But it's on a more advanced level. I would recommend keeping this book inside your mind but start reading later, if you feel very familiar with the basics - even I can't precisely define, what 'basics' mean.

Finally, (iv) is very often cited directly and belongs to the 'canonical' literature to the topics related to the heat kernel. Definitely worth reading - but don't expect do read it from end-to-end, I would recommend you to pick chapters you are immediately interested in, since it contains a wide range of topics.

edit: If you're seriously working on these topics, you'll be faced with functional analysis automatically, it is an important fundament. I also had no 'further' functional analysis education before, but for me it worked learning parts of it 'on the fly' while considering concrete problems. So don't hesitate if you never heard a special lecture. The book if Rosenberg e.g. does also contain very much of useful functional analytic material.

Another worthwhile book that hasn't been mentioned is Liviu Nicolaescu's Lectures on the Geometry of Manifolds which is freely available from his website (under papers'n stuff). While it doesn't cover the Atiyah-Singer Theorem itself, it does a great job of addressing some of the prerequisite material such as elliptic operators (including Dirac operators).