I am very enthusiastic about learning QFT. How much group theory would I need to master? Please could you recommend me a textbook on group theory, which would help me to start QFT?
Georgi's book on Lie Groups is enough, but most of the group theory is explained in the physics texts. It is nice to learn group theory, but the mathematician's theory is more concerned with characters and root lattices, which are nice, but not essential in most of the bread-and-butter applications. The ALE classification is important in mathematical physics, but I think it is covered properly in the physics literature.
You don't need anything too special--- just the rudiments of Lie groups (it doesn't hurt to know group theory, though, it is just not essential). You can learn everything on your own from the QFT source and thinking it out--- there SU(2)/SU(3) cases are not too bad, and these are about as big as it gets. SU(5) and E8 require more sophistication, but are best covered in GUT papers and Green/Schwarz/Witten (for a great introduction to E8)
The modern algebra you probably want to learn is not group theory, but homological algebra, category theory, and Hopf algebra. These are covered well by Lang's algebra book, which is a graduate school staple in mathematics. It doesn't hurt to know everything in Lang--- it's well written, as everything by Lang--- although a little philosophically annoying for me, because it is so conservative in its set-theoretic appratus.
Physics from Symmetry is a book that explains all group theoretical concepts needed to understand the foundations of QFT in great detail and is written specifically for physicists.
It's not very technical, but it's great if you want to understand quickly what concepts are really important for modern physics and why.
For example, it explains why things like the Dirac- and the Klein-Gordan equation or spinors are direct consequences of the Poincare group, which is the set of all tranformations that leave the speed of light invariant.
Actually to start learning the basics of QFT you do not need so much group theory [a different thing is if you want to go to the details]. Some of the introductory books in QFT have at the beginning a section about Lorentz and Poincaré groups, scalar, tensor and spinor representation etc. This is the case, for example in Maggiore, A Modern Introduction to Quantum Field Theory.
If you want group theory for physics for its own sake [which I find useful and beautiful] you may want to learn Lie groups and representations. Start with the link given by Vijay Murthy. There are also very good courses in the internet, I can recommend you some if you want.
Contemporary Abstract Algebra by Joseph Gallian is a good introduction to group theory.