Instead of following the books, I've been teaching group theory for physicists by following these papers below. The idea is to study the papers from top to bottom, and use a traditional books (e.g. Tinkham, Hammermesh, Dresselhaus, Joshi) to fill the gaps.

1. Group Theory and Normal Modes, American Journal of Physics 36, 529 (1968)
2. Nonsymmorphic Symmetries and Their Consequences (unpublished report for a MIT class)

These only cover point group and space group symmetries for solid state physics. For the next semester, I may use also this paper:

3. Galileo and Lorentz Transformations: a study via group theory (in Portuguese)

But it would be nice to complement these with a paper that uses Lie algebras to solve a simple but interesting and illustrative problem (undergrad level). Any suggestions?

From the list of new books listed in the other Answers, I like "Anthony Zee - Group Theory in a Nutshell for Physicists". I'll add to the list these two:

1. A. W. Joshi, Elements of Group Theory for Physicists
2. Zhong-Qi Ma, Group Theory for Physicists