I am an undergraduate non-math major. I just finished my university's engineering calculus series, looking forward to linear algebra in the coming semester. To be frank, I always despised math because I couldn't help but find it tedious. I have recently had a change in heart, however, due to advice and encouragement from my favorite professor and some experience with proving logic statements (which I mysteriously find ridiculously fun…). Starting to read math review journals sounds like a great way to begin as an amateur mathematician.
So where can one find (free, online) mathematical articles with a fighting chance to be understood by high school students and undergraduates?
(The "free, online" part of the question is nonmandatory. I have access to academic journals through my university.)
Best Answer
http://plus.maths.org/ describes itself as "an internet magazine which aims to introduce readers to the beauty and the practical applications of mathematics." It's very well written. The level of the maths varies, but I think you'll find some content that's at the right level for you.
And to echo @Alon Amit's comment - Martin Gardner was a very lucid and entertaining maths writer, and several of his books and magazine columns are knocking around on the web in one form or another.