I second Baby Rudin.
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.