I will say: many areas and not a lot. Ha!

Let me explain: one the one hand, linear algebra and calculus are enough to consider a lot of non-trivial problems and describe basic issues in many areas. On the other hand, the various areas of mathematics tend to interact intensely with each other, which is what makes math so cool. So it's going to be difficult to direct you to a specific area, since chances are that a reference that is advanced enough will not be shy about using much more advanced notions (check out the math articles on wikipedia to get an idea of what I mean; even innocuous sounding ones can get pretty intense).

I do want to encourage you to give in to your curiosity: but instead of picking a specific subject, you would be much better off picking up specific references that are written more specifically for your level. There are many of those, look for general math books, e.g. from the AMS and MAA. "Proofs from THE BOOK" might be a bit intense, but roughly at the right level.

Since the various areas of math tend to riff off each other as I mentioned, the last thing you want to do is get specialized too early anyway, so generalist books are better for you now.