do you just pick a topic/book which takes your fancy, or is there a sorta list you like to follow?
Let's just say there are some books that are a must!
That being said, the similarity of the terms is basically a coincidence. However I have heard of some of the more Mathematical treatments of Feynman diagrams referring to things like "vertex algebras" which my poor understanding takes to be the particular algebra that the field variables obey in specific Feynman diagrams. Does the Mathematics of the field variables create a (Mathematically defined) field? I've never heard of such, but then I can't follow the Math at that level to be able to tell you for sure.
As it was noted before group theory is used in theoretical physics.
Lie groups and algebras are used in classical electrodynamics, quantum electrodynamics and relativity theory (Lorentz group). Group theory is also used in solid state physics for investigating symmetries of crystals.
Algebra can also have more practical application. I heard it is used in cryptography and signal processing (quaternions).
I have found also that link: Group Theory and Physics, maybe it will interest you. I don't have time for reading that because I'm at work recenty