I'd recommend the following three books to similarly cover the three main areas of higher mathematics (analysis, algebra, and topology):
1. Principles of Mathematical Analysis by Rudin (http://www.amazon.com/Principles-Mathematical-Analysis-Inter...)
2. Algebra by Artin (http://www.amazon.com/Algebra-2nd-Featured-Titles-Abstract/d...)
3. Topology by Munkres (http://www.amazon.com/Topology-2nd-James-Munkres/dp/01318162...)
Working through all three (including the exercises!) will give you a solid understanding of the basis of modern mathematics. If you don't have experience with proofwriting, you might find them difficult at first - the activity is very different from performing calculations or solving equations. It's also best to have someone trained in mathematics talk to you about the proofs, until you develop a feel for the needed level of logical rigor.
Rudin, in particular, leaves a lot of work to the reader; going through that book is the most intellectually difficult work I've ever done. If you find it hard-going (which is completely natural), you might want to try Artin first, especially since you have some background in algebra.
(Incidentally, these are the three books used to teach analysis, algebra, and topology to MIT mathematics majors. You can look up the assignments and exams for the three courses - 18.100B, 18.701, and 18.901 - for a good list of exercises to work through.)
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