What next?

I've finished singe variable calculus and I want a taste of pure mathematics before I go to university next September. I'm going to get a grasp of sets and proof first but what next? Number theory, analysis or what? Book recommendations would be ideal!

Mathematicians don't do "we've found a problem which is useful to solve, let's solve it." Instead, they do "we've solved a problem, it seems. Perhaps this can be expanded upon so that it may become useful for other things." In other words, mathematicians work in generalities in that it MAY become useful, whereas engineers don't do problems unless they already know that it is useful.

But other than that, engineers do things that are frequently actually useful for the economy, and permanently useful in that matter. I don't know how the economy could be affected by mathematicians.