OP, I love that your 10yo wants to be a mathematician. You've already received so many good responses! I especially love quark's advice to keep the love of math, the joy, and the curiosity alive. This was always my goal & is far more worthwhile in the long run than any curricular advice you could get.
I identify with being a mathematician, even though it's been a while since I've worked formally. Math was my interest since I was in the third grade and read Fun with Mathematics. Then I went to the library and worked through the cryptography books there. I was blessed with a wonderful high school teacher who introduced me to diverse topics (like AoPS today) and got me started on contest math. In college I got dual degrees in chemical engineering and math, and finally went on to get a PhD in applied math.
I've worked in academia (Ga Tech), industry (duPont central research office), and high school teaching & tutoring (CTY and others) over the years. Each had pluses and minuses. Academia was great in the freedom allowed to pursue my own interests (nonlinear elastic material science & population dynamics), and I loved teaching bright young students, but the stress of publishing and gaining tenure was bad, especially when I wanted to get married and start a family. DuPont paid much better, and I got to work on all sort of real life problems, from designing toothbrush bristles to airplane wings. CTY allowed me to interact with some really sharp young minds. All of them offered hard problems and great co-workers.
I agree that national labs are another wonderful place for a mathematician to consider. I met several NSA mathematicians when my dd was on a Maryland based ARML team (they came to some of our practice meets), and I considered taking a position at Sandia Labs in New Mexico back in the day.
My son is a former USAMO kid & MOPer, who's enjoying work as a software developer at one of the Silicon Valley companies.
My daughter loves putting her math talent to work at the AoPS offices, helping to run their online school. Plenty of past math olympiad kids work there.
Both kids found lots of peers through summer math camps, AoPS classes, and math competitions.