I'm planning on taking Abstract Algebra and Real Analysis in the future, but I realize I am very lacking in "mathematical maturity". Having only taken a computational linear algebra class (Anton), would a good next step be to do Axler's Linear Algebra Done Right?
Additionally, does Hubbard's vector calculus, linear algebra, and differential forms prepare you well for Artin's Abstract Algebra and Baby Rudin?
Thanks!
On
As someone who has read Axler's book in first year, the questions can be challenging for people who have never been exposed to proofs. With that said, pairing yourself up with Book of Proof by Richard Hammack, How to Prove It by Daniel J. Velleman (as suggested by Coy Cattret), or How to Solve It by G. Polya might be better before taking a dive into serious mathematics.
Of course, this is also not meant to discourage you from learning from Axler as he is a phenomenal author and his proofs are concise, "simple" and (like all mathematics) build upon themselves.
The first thing I'd like to say is that some students require no real preparation for rigorous math. As soon as they start reading introductory algebra or analysis, they immediately "get" proofs. Others struggle, though, because they are confronting two problems at once: learning to write proofs and learning unfamiliar concepts at the same time. The only way you'll find out how hard it is for you is by starting!
By the way, Artin and Rudin are both good choices. Artin is a bit easier than Rudin, so you could start with that. An alternative to Rudin that starts more gently but actualy ends up covering more (and more interesting) material is Apostol. If you have any concerns about your ability to deal with proofs, you could choose Apostol instead of Rudin and lose very little.
Either way, working through Eccles' book should definitely provide sufficient preparation. An alternative is Journey Into Mathematics: An Introduction to Proofs by Rotman, which has quite a different philosophy to Eccles. Rotman doesn't focus as much on logic and foundations, but instead discusses substantive mathematics that is appealing in its own right. That being said, the least interesting parts of Eccles are already behind you now.
Although I have a general idea of the content in Hubbard, I've never looked closely at its problem sets, so I can't answer your last question. In an analysis or abstract algebra course, problems are often at the same level of difficulty as proving some of the theorems in the text (of low or medium difficulty, not usually the hardest theorems). If enough problems in Hubbard are at this level, then it ought to prepare you.
If you're not in a huge rush to learn multivariable calculus (say because you have a class you need it for or because you need it for physics), I would probably recommend delaying it until you've learned basic analysis. That way you'll be able to approach it from a much more advanced standpoint.
Also, there will be no need to read Axler, as Artin covers everything you need in linear algebra, mostly in the first half of the book.