I am trying to self-study abstract/modern algebra. I am currently trying to learn from Josephn Gallian's Contemporary Abstract Algebra 7th edition and free lecture notes from the internet.
So are there any other good textbooks for reference? Or anything better than Gallian's?
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.