Suggestion for modern algebra textbook 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.