A good place to start is in low dimensions and there a good introduction can be found in
J. Kock, 2003, Frobenius Algebras and 2-D Topological Quantum Field Theories, number 59 in London Mathematical Society Student Texts, Cambridge U.P., Cambridge.
There are also old notes of Quinn that take a very neat homotopy theoretic approach to some of the problems:
F. Quinn, 1995, Lectures on axiomatic topological quantum field theory, in D. Freed and K. Uhlenbeck, eds., Geometry and Quantum Field Theory, volume 1 of IAS/Park City Math- ematics Series, AMS/IAS,.
These do not really get near the physics but you seem to indicate that that is not the direction you want to go in.
I like the links with higher category theory. I realise that this is not everyone's `cup of tea' but it does have some useful insights. I wrote a set of notes for a workshop in Lisbon in 2011 which contain a lot that might be useful (or might not!). They are available at
http://ncatlab.org/nlab/files/HQFT-XMenagerie.pdf
My advice would be to raid the net getting this sort of resource (storing it on your hard disc rather than printing it all out!), then as you start working your way through some of the stuff you have found there will be explanations available ready at hand. Start with the main ideas and `back fill', i.e. don't try to learn everything you might need before you start. If when reading some source material an idea that you are not happy with comes up, search it out then, just enough to make progress beyond that point easy. (Of course this is how one progresses through lots of areas of maths so ....)
(Those notes of mine exist in several different forms and lengths, so in a longer version some idea may be more developed.... so ask!)
.... and don't forget the summaries in the n-Lab can be a very useful place to start a search.
(I had at the back of my mind just now a reference to a seminar that I had some notes on .... but no idea of the source. A little search found me:
http://ncatlab.org/nlab/show/UC+Riverside+Seminar+on+Cobordism+and+Topological+Field+Theories
... One other thing, there are lectures by Lurie on this stuff on YouTube:
https://www.youtube.com/watch?v=Bo8GNfN-Xn4
that are well worth watching.)
Concerning advanced differential geometry textbooks in general:
There's a kind of a contradiction between "advanced" and "textbook". By definition, a textbook is what you read to reach an advanced level. A really advanced DG book is typically a monograph because advanced books are at the research level, which is very specialized. Anyway, these are my suggestions for DG books which are on the boundary between "textbook" and "advanced". (These are in chronological order of first editions.)
- Bishop/Crittenden, "Geometry of manifolds" (1964). Quite advanced, although not too difficult, despite the 1964 date.
- Cheeger/Ebin, "Comparison theorems in Riemannian geometry" (1975). This is on the boundary between textbook and monograph. Definitely advanced, despite the 1975 date.
- Greene/Wu, "Function theory on manifolds which possess a pole" (1979). Monograph/textbook about function theory on Cartan-Hadamard manifolds, including extensive coverage of Kähler manifolds.
- Schoen/Yau, "Lectures on Differential Geometry" (1994). This is about as advanced as it gets. You need to read at least 5 other DG books before starting this one.
- Theodore Frankel, "The geometry of physics: An introduction" (1997, 1999, 2001, 2011). This has lots of advanced DG, but in the physics applications, not so much on topological DG questions.
- Peter Petersen, "Riemannian geometry" (1998, 2006). Very definitely advanced. You need to read at least 3 other DG books before this one.
- Serge Lang, "Fundamentals of differential geometry" (1999). This is definitely advanced, although it nominally starts at the beginning. It's what I call a "higher viewpoint" on DG. Very thorough and demanding.
- Morgan/Tián, "Ricci flow and the Poincaré conjecture" (2007). Advanced monograph on the Poincaré conjecture solution, but written almost like a textbook.
- Shlomo Sternberg, "Curvature in mathematics and physics" (2012). Definitely advanced. On the boundary between DG and physics.
I would say that all of these books are beyond the John M. Lee and Do Carmo textbook level.
Best Answer
I came across a nice video lecture by Niyogi that gives a nice survey of manifold learning. I thought I would share in case anyone else was interested.
http://videolectures.net/mlss09us_niyogi_belkin_gmml/