What fields should I focus on? First of all, a little background: I'm a computer engineering student, currently on my second year of study. I've always been fond of mathematics but my interest has grown a lot since last year, mainly because of my first formal take on it at my mathematical analysis class.
Since then, I've been considering switching to a math major but I still have my doubts. Anyway, I decided to start learning maths on my own as well. I've taken a mathematical analysis (proof oriented, single-variable calculus) course, a linear algebra course (once again, proof oriented) and I'm taking a multivariable calculus course right now. I'm working on my own on abstract algebra (using Herstein's Topics in Algebra) and introductory topology (just started with Topology Without Tears by Morris).
I may remark that studying with textbooks on my own is considerably harder than taking a course; I seem to be progressing really slowly with those (I try to do every exercise though), compared to my performance inside a course, where I normally don't have problems at all.
So basically, my question is a bit broad: What do you think I should focus on? I'm mostly interested in pure math, since I have a few applied math courses later on my major. Book recommendations and general advice are welcome as well.
PS: English isn't my first language and I realize the above is a bit rusty, feel free to edit it if you find errors.
 A: Read up on the subjects which intrigue you and which you find beautiful and enjoyable to think about.
A: Have you considered adding math as a major? Or would that be unrealistic to manage, in addition to working toward your computer engineering?  I am not discouraging you, in the least, for your pursuit of math via self-study, but if you're putting in all the time and effort on your own, why not consider putting that time and effort (and passion!) towards earning the second major?  
Perhaps you can discuss such a prospect with a favorite mathematics professor you've had?  At the very least, there may be some classes that you can take, that will not only contribute toward your math-learning, provide a bit more structure, but also give you the opportunity to get acquainted with other students passionate about math.  Personally, I've studies quite extensively, on my own, and while it's rewarding (and encouraged), there is also something to be said for "after hours" study groups with fellow students, socializing with those sharing your interest, etc.  Self-study can be a bit lonely at times, unless you create avenues for sharing your work and interest with others...
(Of course, you're always welcome to join our community, here at math.SE!)  :)
Also, I concur that what is often key is not getting too worried, at this stage, about what one "should" study.  It's a good question, don't get me wrong.  But along with some of the "should takes", be sure to add some "want to take/study"!
It's not just a matter of balancing "work" with "play"; it's sort of like combining work with play!  
