If you have zero familiarity with representation theory you probably want to start from something like
Theory of Group Representations by Naimark and Stern, that contains representation of finite groups, compact groups and some Lie theory.
The most common source of irreducible infinite dimensional representations are induced representations, for which I recommend
A Course in Abstract Harmonic Analysis by Folland. This book also contains basic C* algebras stuff, representation of locally compact abelian groups (i.e. the closest generalization of Fourier theory), representation of compact groups and a treatment of induced representations via the "Mackey machine".
If you are interested in applications, induced representations are used in the foundations of quantum field theory. If you have some familiarity with quantum mechanics you may find the new
Quantum Theory, Groups and Representations by Woit, to be a fun reading while working through the more mathematical books above.