The short answer to your question is yes. Things are proceeding normally in your undergraduate mathematics education.
My math studies were typical and looked like this:
- Calc I - IV
- Foundations (Logic, Set Theory, Proof Methods)
- Modern Geometry
- Probability and Statistics
- Linear Algebra
- Abstract (Modern) Algebra
- Real Analysis
- Discrete Math
- Electives, Independent Studies, etc
My favorite courses were Abstract Algebra and Topology. Also there was a bunch of computer science in there. Your mileage will vary. TL;DR whatever.
Here is an excellent video on the history of mathematical thought. In just 50 minutes Dersch touches on about 5,000 years of material. Your education will parallel his exposition, sort-of.
Anyway, different programs at different schools might assign prerequisites among these as they please to suit their curriculum, but generally you will find that the "arrows" point downward (e.g. Calc is a prereq for the Algebras).
What I call Foundations above is the rite of passage of mathematics - it transitions you out of childhood and into adulthood. I know you're confused by these sudden and unexpected changes. It's OK, grasshopper, let it happen, and when it is done you will be a man.
You see, pure math is about what is true, how we know it is true, and how to make sense of it all. In high school you learned to find the roots of a quadratic polynomial, but in college you will learn why it is impossible to do the same thing for a polynomial in the 5th or higher degree.
As for lots and lots of calculation, well, you will be doing lots and lots of calculation, but it won't be numbers as such. Check out the famous problem of the Seven Bridges of Königsberg. There are no numbers involved (except seven, which is important), but to find the answer you must do computation in some sense. It is that sense in which things get interesting in higher mathematics. A once girlfriend of mine once pulled a math book off my shelf and said "there are no numbers". That means that it is a true math book, that you are a man.
If you really want to just learn to calculate stuff and solve real problems, applied mathematics is better for you. At Harvard, Bill Gates declared as an Applied Math major because it allowed him to take courses from all over the college that were otherwise separated from each other: economics, business, science, nothing was off-limits.
Math is the study of what we know, Applied Math is the study of how to apply it to the real world. For example, in Linear Algebra you will learn the theory of vector spaces, but in Applied Mathematics you will use vector spaces to model the human face.
The choice is yours. If you want to understand the universe and be smarter than everybody in the room and have them hate you because you're right about everything all the time, or if you want to teach for the rest of your life, then study pure math. If you want to make eigenface generators or do hyper-fast quantitative Forex trading or build shoe-computers to beat the roulette wheel in Vegas, then you should study Applied Math, or you could just completely douche out and study Physics.
It is a vast world, but it is modern; you conquer it with mathematics. Good luck.