Mathematics is incredibly broad, but that's not very apparent in what's taught in school. One of the best ways for students interested in math to learn more about it is to attend summer math camps. Just as importantly, those camps provide a community of math-loving people who usually share lots of other interests, as well. Some of my best friends are people I met at math camp.
Here, I'll go through some summer math programs that focus on theoretical or pure math as opposed to problem solving math. One, MathPath, is for middle schoolers. The others are all high school programs. There's a little more detail about Mathcamp social life because I'm a Mathcamp alum, so I know more, but all of the programs have excellent community aspects. All of the websites have much more information than I could list here, including daily schedules and course offerings from previous years.
