Baby Rudin is a great choice, especially for self study, since there are a number of study guides for it that go through solutions, additional explanation of material, etc. I'd recommend against the chapters on differential forms, though, just because the treatment is outdated. Marsden and Hoffman's approach is to focus on explanations for why theorems are true before giving proofs, which some people find useful.
Axler is a good choice for linear algebra. Dummit and Foote is the standard choice for algebra generally. I'm of the opinion that we should teach algebra before linear algebra in general, but this seems like a minority view.
The problem sets are nearly legendary and the writing is terse. I might have to read through it again myself sometime.
"Algebra" by Artin is also a great choice if you want to learn about modern algebra. There's some good stuff in there for looking at linear transformations in, say, 2-space, as groups.
Axler is a good choice for linear algebra. Dummit and Foote is the standard choice for algebra generally. I'm of the opinion that we should teach algebra before linear algebra in general, but this seems like a minority view.