Is first year math in an university just Calculus I, II (or AP Calculus BC), and III?
It depends on the strand. Some schools only have one strand but a lot have a standard Calc I and II strand and then others do an honours version of this.
The main difference is that you will be exposed moreso to more advanced topics like the proofs of the various identities and so forth and the level of abstraction will possibly be higher. It's kind of like the difference between studying generalizing geometry with manifolds as compared to studying geometry in Euclidean (note Cartesian) space.
You could look at Harvard Maths department and that should give you a rough idea of some differences that could be encountered. Also note that they have one of the hardest undergrad courses in math which is the Math 55.
Calc I and II is usually assumes that you've done on limits and calculus because it goes at a decent speed and you will have to do your work to keep up. You'll touch things like some basic differential equations as well as series sequences and other things. The lower levels is about getting used to doing simple calc problems so that you can use that to build on doing harder problems when you learn more advanced techniques with differential equations and other advancements in calculus.
Calc III builds on geometry aspects that you study in first year. It can be done in the first year when its taught as an multivariable calculus course but it expects that people have already done into calc and have gotten good results.
Also you will learn basic linear algebra.
Most of what you learn in the early days of math is simply a lot of techniques. Its kind of like building a collection of tools to tackle problems. Although there are stark differences between honors and normal, if you're going to do more applied math the tools that you use (as long you understand them) should really suffice.
Good luck with uni when you get there
So it is not...?
Separate names with a comma.