Reddit reviews Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy (Dover Books on Chemistry)

Cotton's "Chemical Applications to Group Theory" is pretty much the basis for all undergraduate classes that teach group theory. It's expensive though, and probably not the first book you'll want to read on the subject.

I would recommend Bertolucci's "Symmetry and Spectroscopy". It has a lot of great info, and is only $15.

Some good online sources (not all notes are about group theory, so pick and choose what will help you):

http://ocw.mit.edu/courses/chemistry/5-04-principles-of-inorganic-chemistry-ii-fall-2008/lecture-notes/
http://chemistry.caltech.edu/courses/ch112/syllabus.html
Under "Symmetry in Chemistry"

You should also have a working knowledge of matrix algebra. If you want to look into the subject deeper, a good understanding of linear algebra will help.

I highly recommend this book:

https://www.amazon.com/Symmetry-Spectroscopy-Introduction-Vibrational-Electronic/dp/048666144X

My inorganic professor recommended it when I took the course and it was a great investment.

Have a look at this book.

http://www.amazon.com/Symmetry-Spectroscopy-Introduction-Vibrational-Electronic/dp/048666144X

This book does a great job explaining the methods for predicting active vibrations in IR and also has great stuff on electronic absorption spectroscopy.

A good upper-level undergraduate textbook with plenty of practice problems is Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy by Daniel C. Harris and Michael D. Bertolucci. This book is pretty thorough in its explanations, so if you work through it start to finish, it may help you better grasp some areas that are currently not clear.