There is a new book called Physics From Symmetry which is written specifically for physicists and includes a long, very illustrative introduction to group theory. I especially liked that here concepts like representation or Lie algebra aren't only defined, but motivated and explained in terms that physicists understand. Plus no concepts are introduced which aren't needed for physics, which was always a big problem for me when I read books for mathematicians. Group theory is a very big subject and mathematicians find a lot of things interesting that aren't very relevant for physicists.
Although if you're looking for mathematical rigor, this may be the wrong book and I would recommend Naive Lie Theory by Stillwell.
In fact, my recommondation would be to read both. The first one to understand what concepts are important for physics and to get a first idea for the motivation behind them and then Stillwell's book in order to get an idea how mathematicians think about these subjects.