In What order should I Learn math in?

I recommend finding some old math tournament problems, there's a million to chose from. Find some at about your level and learn whatever you have to do in order to be able to solve the problems.

If there is a question that interests you that you couldn't solve, figure out what you need to study in order to be able to solve it.

There's a world of difference between "knowing" an area of math, and being able to use it like a weapon.

I think, at your age, you can play more with Combinatorics, planar geometry and elementary number theory. These fields are full of interesting and exciting results and most of the time, these results have interesting history which gives information about history of mathematics.

It is always good if you study mathematics in a formal way starting from formalism of set theory. But math is not only about the precision and formalism, it is also about having vision and intuition.

I believe that starting with basic combinatorics, geometry, number theory and analysis is a very good way to develop a preliminary vision about what is mathematics and what it is going to do.

Also, solve as many problems as you can!

artofproblemsolving.com is a website where students learn math with an emphasis on creative problem solving, and also train for math contests. They have developed a curriculum, including books on number theory and counting and probability. The curriculum emphasizes combinatorics, planar geometry, and elementary number theory, as Arash suggested.