I'm a math major. I took a three-quarter sequence of Honors Calculus, which was probably unlike the calculus classes at almost every other school in the nation. We never did volumes of rotation, double integrals, anything like that. Lots of delta-epsilon proofs and axioms and definitions. I think I computed like 6 integrals in the entire course.
The delta-epsilon definition of a limit: The limit of f(x) as x-->x_0 equals F <==> For all epsilon>0, there exists delta>0 such that if (x-x_0)<delta then (f(x)-F)<epsilon. I still remember that, even though it was three years ago that I learned it, and I haven't used it since.

Most textbooks and college courses state this definition at the beginning of the course and quickly forget about it, AFAIK. That's definitely what happened in my AP Calculus course in HS.
Last year I took a three-quarter sequence in Real Analysis. That was pretty hard for me. It's difficult for me to think in terms of the infinite.
This year I'm in a three-quarter sequence of Abstract Algebra. The Group Theory stuff was all very cool and easy for me. Once we started moving towards Linear Algebra, though, it got pretty hard. But again, I think that algebra here is different from anywhere else. As with all math here, it's a lot more theory-based and a lot less algorithm-based than at other places. At most schools, math is seen as a means to an end: engineering, physics, whatever. And that's how it's tought. Here, math is an end unto itself. In my year-long analysis course, we didn't learn a thing about solving equations with Legrange Multipliers. I learned that in Econ. In my Algebra course, we're now moving into Galois Theory, and it's pretty mind-blowing.
Definitely take a course on Group Theory, Rings, Fields, etc.... it'll expand your conception of the world. It'll teach you to think outside of our realm of 'numbers' and force you to re-think what numbers even are (arbitrary products of our imagination that work only in accordance with set rules, and it's the rules that really matter, not the numbers).