There is no easy or right answer. I know brilliant professors who cannot easily decide what textbook to use for an advanced math course, and for good reason. Every book has its own strengths and weaknesses. I suggest you go to your math library (assuming one is available during this pandemic) and examine several books. A book you like might be hated by someone else, it is highly individual. You likely will need at least two or three books so you can go back and forth. Even a good book can be bad in a particular section and vice versa. Use a common textbook that has gone through at last two or three editions as a guide as to what topics to cover and then be prepared to use alternate books to actually learn the topic.
Use the Internet. Don't be afraid to read lecture notes or check Wikipedia. Also, Professor Keith Conrad (Univ. of Conn.) has dozens of expository papers on algebra on his web site, some are easy, some are difficult, and some are advanced or specialized. I have found that lectures by professors at lessor known universities to often be better than those by professors at famous brand name universities. That being said, I have found lectures by Unv. of Berkeley professors to be quite good, and lectures by MIT professors to also be good, but the latter are often very fast paced and better for review than to self learn from as they are so intense.
I suggest you get an easy book, an intermediate book and eventually a hard book. Herstein's: Topics in Algebra is harder than Birkhoff and MacLane's book, but Birkhoff and MacLane's book is good for learning the fundamentals. As an undergraduate I used Herstein, but I think it is too difficult to self study from.
It is critical to learn the definitions and other fundamentals cold and then go on to a more advanced treatment. (One really smart professor basically told me: memorize definitions, but do not memorize proofs, just understand them.) Herstein loves to give problems and results that are hard using elementary methods, but easy using more advanced methods. In my opinion this a bad way to learn, as not everybody is clever at solving hard problems or following highly technical arguments, and I think it is more useful to put one's energy into learning the concepts and theory that makes it possible to eventually easily understand what is really going on, rather than rely on clever technical tricks or manipulations to get a result with no real deep understanding as to what is really going on. Neither Herstein nor Birkhoff and MacLane cover everything a graduate course would cover. Herstein, in my opinion, makes the subject seem more difficult than it is.
A free book is by Robert B. Ash (University of Illinois at Urbana-Champaign), titled: Abstract Algebra: The Basic Graduate Year, it is available as a series of PDF's on his web site.
I also use: Algebra: A Graduate Course by I. Martin Isaacs, it has its strengths and weaknesses. It is is elegant and the proofs are carefully done, but it may be too abstract and condensed to self study from.
A classic is the two volume (mostly of the time only the first volume is used) set by B. L. van der Waerden titled: Modern Algebra. There is at least two English editions (the original is in German). Even though the editions differ, any English edition is fine. And if you really groove on abstraction there is Serge Lang's book, simply titled: Algebra.
Alternatively, audit the class you need. Self study is a mixed blessing, as while there are no exams, required homework or time pressure, it can be frustrating and inefficient.
Good luck. I think it is great you are so motivated. Keep the faith. Don't worry if at times you get overwhelmed or discouraged --- self study is not easy --- it has happened to many of us at some point in time, yet somehow we didn't let it stop us, nor should you.
