I highly recommend Michael Artin's Algebra along with these course materials. You can also find supplementary materials (exercises, solutions, & notes) here.
The other canonical choice is usually Herstein's Topics in Algebra.
A book similar to Artin, but more leisurely, is Vinberg's A Course in Algebra.
For group theory, I have really been enjoying the first 100 pages of Rotman's An Introduction to the Theory of Groups. It is quite readable!
And if you know a bit of algebra already, you might try Aluffi's Algebra Chapter 0. It is nonstandard (some use of categories from the start), but a pleasure to read. Plus, it's free!
If you're only looking for a reference, then I would have to say Dummit & Foote, as well. But as a main text... meh. But there are solutions for many exercises here.