the dover books are also a good series, and pretty much anything by Artin is good. i also looked at the Halmos book a couple of people have mentioned.
perhaps one thing to be aware of, though, is that you're not always going to learn things in the linear way they're layed out on the page. moreover, it might be helpful to have more than one book for any given subject. Lang's Algebra, for example, is a really good reference, but a tome if you read it like a text. so you might pick up something small and subject-oriented with a lot of exercises like Artin's Galois Theory or Atiyah's Commutative Algebra, and supplement it with a reference like Dummit and Foote or Lang's texts on Algebra as a whole.
oh, right: whatever book you choose, do the exercises.
But that's not a Springer book, it appears to be published by Dover, and the covers look totally different. The GTM series is also highly variable in quality.
perhaps the GTM series has some bad titles; i haven't read them all. but it has some good titles, and from the description original poster's background, it's probably as far as the person wants to stretch right now. sure, the cambridge advanced math series might be more "quality," but those also tend to be both very focused (in terms of specificity of the topic) and very dense (in terms of delivering a lot of results with almost zero fluff).
replace "that" with "those" ( http://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics ).
the dover books are also a good series, and pretty much anything by Artin is good. i also looked at the Halmos book a couple of people have mentioned.
perhaps one thing to be aware of, though, is that you're not always going to learn things in the linear way they're layed out on the page. moreover, it might be helpful to have more than one book for any given subject. Lang's Algebra, for example, is a really good reference, but a tome if you read it like a text. so you might pick up something small and subject-oriented with a lot of exercises like Artin's Galois Theory or Atiyah's Commutative Algebra, and supplement it with a reference like Dummit and Foote or Lang's texts on Algebra as a whole.
oh, right: whatever book you choose, do the exercises.