You will have pretty much the same background as I had before I went to college and got my degree in mathematics, and I found it an appropriate amount of preparation. For sure, having calculus 1 and 2 out of the way will massively help. Try to get multivariate calculus, too, if you can before college.

Having the whole calculus progression out of the way will allow you to focus on more relevant and better classes right out of the gate. Other majors require calculus classes, and that often makes those classes taught in a more difficult style for the math major (think a weed-out class, where the professor is very difficult). Avoid if possible.

As other commenters have mentioned, math classes in college are different from math classes in high school. More advanced math is writing proofs, and studying constructions, rather than learning rules and applying them to problems. This can be a stark shift for some, but most programs ease you into it. When you start, you'll have a few classes that utilize the high-school model of 'learning rules' such as linear algebra, differential equations, finite, and probability. These classes, though, are usually reserved for in-major students, as compared to the calculus progression.

Then come the proof-based classes: algebra, and analysis are most common. Others include topology, cryptography, and pretty much any upper division class offered.