I am also a mathematics student who likes physics. My advice would be:
- I learned a bit of QFT following Srednicki's book and Tong's lecture notes. A note: the book focuses mainly on the path integral approach, with very little in the way of S-matrix techniques.
- For GR, I found Jetzer's lecture notes and, knowing the guy, Graf's lecture notes ought to be great.
- I have learned String theory on some book, but I don't remember the title right now. If you want can look it up.
I think here at ETH there's a pretty good group doing mathematical physics, if you want to try to apply and to move to Switzerland.
Also general advice: If you are interested in a topic in physics, always look if Tong has written something on it. His lecture notes are usually very good.
This got too long for a comment but is meant to be an extended one. I'm note quite the guy to say being mostly interested in it from a structural/mathematical perspective. Forgive me if I'm not telling you anything new.
You can definitely do TQFT within the confines of pure math. If what you want is the standard model you'll do well to understand your representation theory, as types of particles correspond to fundamental representations of Lie groups ($U(1)\times SU(2)\times SU(3)$ in the standard model, times the Poincaré group if you do the analysis.) From there a quantum field is a section of a vector bundle associated to the representation over space-time satisfying a variational principle (an extremal of an action) involving suitably equivariant connections (which are incidentally your bosons). Faria-Melo develops this and in fact exhibits the standard model in this framework.
They leave out a clear analysis of how representations tie in with types of particles, but this is done by Baez and Huerta in this text (http://math.ucr.edu/~huerta/guts/). Basically, elements in your fundamental representations are fermionic particle states, generators of the adjoint representation are bosons that act on your fermions in a way that can be represented by Feynmann diagrams.
Quantization is still fluffy to me, but it appears this is where quantum groups come in: You cannot deform a semi-simple Lie algebra and get a reasonable deformation of its representation theory (it's category of representations). You can however deform its universal enveloping algebra (which is a Hopf algebra, i.e., an object with a favourably interacting product and coproduct). There is a master class on this going on right now which talks about this for the purpose of studying 3-manifold invariants using 3-dimensional field theories. Notes about quantum groups may be found on its web page: http://www.math.ku.dk/english/research/conferences/2014/tqft/ They have incidentally a crash course on operator algebras as well, which is part of the theory that allows you to reasonably deal with infinite dimensional representations of the Poincaré group.
How the functor point of view on field theories relate to the "classical" one developed in Faria-Melo a bit fuzzy to me, but I suspect you may find some answers in Segal's article on conformal field theories (http://www.math.upenn.edu/~blockj/scfts/segal.pdf -- a pretty shitty scan but you'll find it in his 60th birth day thing).
Of course this leaves out nitty-gritty computational aspects of the kind a physicists would be able to tell you about, and I have never gotten close enough to what the physicists do to actually wanting to renormalize anything (something you apparently need to do because of self-interacting particles producing diverging integrals). This is definitely a pretty big part of QFT you'll be missing if you don't study the physicists approach as well.
It appears the big unifying idea in any case is that a physical system should be invariant under choice of presentation (gauge) up to a group or automorphisms (gauge transformation) and that this is true for classical systems (Lorentz or Poincaré invariance of space or space-time) as well as quantum systems (other Lie groups acting on a vector bundle of states) and that all of physics are more or less fall out as properties of stuff with the right symmetries. This appears to be what physicists and mathematicians agree on either way, so you can't go wrong studying representations.
Aside from Faria-Melo here are some notes I like to look at:
These notes are pretty explicit about the kind of mathematics they use math.lsa.umich.edu/~idolga/physicsbook.pdf
These notes on Lie groups and representation theory are very good. staff.science.uu.nl/~ban00101/lie2012/lie2010.pdf They come with video lectures. webmovies.science.uu.nl/WISM414
Best Answer
I am a physicist and I had specialized in Astrophysics back at university. Since you already have a background in differential geometry and topology, I would suggest the (one of the) two all-time classics about gravitation:
Note, however, that these are ancient books (mid 1970's). But at least at around 2000 they were still state-of-the-art in teaching. Maybe you can find more recent books. Of course, Gravitation theory as such has not changed since Einstein, but applications and probably some solution techniques probably will. And, of course, our understanding of the large scale structure of the universe has changed a lot since these times. However, dark matter and dark energy, which play a role in the latter are not at all understood, so as far as we know, gravitation theory/general relativity is still the same and only the matter that fills the universe has to be 'corrected' (presumably).
If you want to dive into supersymmetric/string theories and quantum gravitation, I cannot recommend any books because I am not into that. I would not recommend that you start with that because it would probably not give you much physical understanding (which seems to be what you're after), and it requires a lot of quantum field theoretic knowledge as well. On the other hand, you as a mathematician might enjoy the 'otherworldliness' of these theories very much. Be prepared that this has not (yet) much to do with observable physics.
If you want to understand star formation or the so-called 'big bang', there is a lot to understand from quantum field theory and especially nuclear physics. I think this is a huge topic, and you will either spend a lifetime on that alone, or you will end up reading more 'phenomenological' treatments of it.
A word of caution: physics, as it is taught, is a very different style compared to mathematics. I am seeing this from the opposite perspective than you do. While mathematicians lay out their assumptions very carefully and then prove everything, physicists often omit (presumably) self-evident assumptions and only sketch proofs. Sidenote: the most ironic thing about contemporary physics is that quantum field theory is the best working theory we ever had (predicting measurements up to 10 or so significant figures), but it still does not have a rigorous mathematical foundation. This is already painful for me, but I can hardly imagine how this must feel for a mathematician.
This is all for the purpose of getting straight to the point instead of (presumably) wasting time with trivialities. So compared to mathematics, you will need a very good memory of what was said on page 20 while you're at page 500. Most things will be referenced back, but many things will just be assumed that you remember them. And while imagination is kind of 'sugar coating' in mathematics, it is essential (with respect to fundamental physical experiments) in physics. I would even say this is rule #1: know the essential experiments/observations that led to the theories. For gravitation, this is just a handful (Basics of Electrodynamics and Michelson-Morley for special relativity, Kepler's Law, Free Fall, perihelion shift of mercury, Pound-Rebka, gravitational lensing, etc. for general relativity).
It will probably be pretty painful for you to read a physics book as it is for me to read a mathematics book. If I read a math book, I am always impatient with all the abundant notation, the lemmas and auxilliary theorems that are just building blocks for the central theorems. Physics tries to get to the point faster. You will probably find that the way theoretical physics is treated is unbearably sloppy.
Having said that, the book by Weinberg above is a little more rigorous mathematically, as far as I remember.
Of course, your mileage may vary. If you can let yourself go, theoretical physics will not be a problem for you.