Have you ever been to the Mathematics Institute in Leiden? You are always welcome. I will show you around (including the library :)).
As far as I understand, in principle, you can attend some lectures even in the coming year. I would advise the Algebra course given by Hendrik Lenstra. The course material is available in Dutch at http://websites.math.leidenuniv.nl/algebra/
In my opinion, you should not concentrate so much on "doing things properly", but rather read and think about some nice elementary (which is not synonymous to easy) mathematical problem. There are quite a few such problems. In my last year in high school, I was attending a "course" on Pell's equation, which was really nice and served the purpose of doing something outside the standard school curriculum.
Anyway, do not hesitate to contact me.