I'm a Chem. Eng. grad. I did well in my math courses and I really like math. I've started digging for some more to learn on my own. I like applying math but I'm most satisfied when I have a deep understanding of things and a good intuition about it. Given that I have completed 3 calc. courses including multi-variable calc., a course in diff. equations, and linear algebra, what would be good topics to learn next? I'm interested in everything. I especially enjoy trying to prove things myself. I'm intrigued by abstract algebra and complex analysis but is there something I should learn first before jumping to those subjects? Thanks.
One recommendation is to learn how to do proofs (this will be a huge help in all math courses).
Also, you might want the answers here: Strategy to improve own knowledge in certain topics?
If you are interested in the subject in general for its own sake, a wonderful book is Aigner and Ziegler's "Proofs from THE BOOK". The books by William Dunham, e.g. "The mathematical universe", "Journey through genius" are outstanding.
If you want to learn stuff to apply it, go trawl through Open Course Ware, take a look at Coursera. Look at the books from The Trillia Group and the ones they link. The lecture notes by William Chen are very readable. I'm sure searching for "lecture notes" and your specific interest(s) will turn up many more good resources.