I have skimmed Carter's Visual Group Theory and it looks fantastic. A good choice if you have a lot of time, and want a thorough introduction to the basics with

I have skimmed Carter's Visual Group Theory and it looks fantastic. A good choice if you have a lot of time, and want a thorough introduction to the basics with lots of intuition-building and examples. If you want something more traditional -- sparse and fairly terse, largely theorem-proof based, and with most of the action in the exercises -- Saracino's Abstract Algebra is my favorite in this genre.