If you are going to study computer science, you want to read Graham, Knuth, Patashnik: Concrete Mathematics.
Look into solving recursion, combinatorics and maybe a little stochastics. Getting the hang of infinite sequences and series is also advisable. Above stated book covers that all because it is made for computer scientists. You should have math courses covering all that, though.
I am someone who did just as you are hoping to do -- I don't enjoy studying material without a strong background in the assumed prerequisites, so some would feel that I'm suggesting too much.
My suggestion is to go through Stewart's calculus front-to-back, doing typical problem sets. Whenever you struggle with a pre-calculus or algebra fact, make sure you review that material. Once you've done this, I recommend deviating from the typical routine and picking up a copy of the book "How to Prove it." With the maturity you have after going through calculus, you will understand the essence of what a proof is, and the importance of working with and thinking in terms of definitions. After this, for fun you might enjoy going through "Geometry: Seeing, Doing, Understanding" by Jacobs. Then, quickly, go back through your Algebra/Pre-Calculus textbook examples and work through any of them that do not seem immediately obvious how to do. Then, quickly go back through Stewart, doing the same thing. You will be very prepared at this point to do very well in your future studies.
From here, you would want Linear Algebra, Complex Analysis, Abstract Algebra, and Real Analysis. For Linear Algebra, I recommend Gilbert Strang's MIT opencourseware videos, and for Real Analysis, I recommend Francis Su on YouTube (he goes through the textbook Principles of Real Analysis by Rudin).
These, I think are essential. you can't plan too much further in advance because your interests will change as time goes on. But some other areas you could look at are probability and statistics, number theory, numerical analysis, and differential equations, or more advanced topics in what I've already mentioned.
Best Answer
Concrete Mathematics: A Foundation for Computer Science, By Donald Knuth himself!