As a math major in my senior year, I would say that you'll have plenty of time to study subjects such as Abstract Algebra and Analysis.
If I could go back to my last year in high school, I would invest more time into gaining and shaping mathematical reasoning than dutifully study certain specific subjects.
I would go through these books:
How to Prove it by Velleman: Imagine yourself as a soccer player. This book serves like those drills on short passes and ball control. They are the basics, and when you become good you'll do them automatically, but you will only do them automatically if you had first spent several hours practicing the basics.
The art and craft of problem solving by Paul Zeitz. Challenging. The former book focused on the basic logical steps that constitute a mathematical proof. This one focuses on the ideas, and on being creative. It covers the basics of Number Theory, Geometry and Combinatorics and suggests wonderful references that I suggest you also look up.
Discrete Mathematics by Laszlo Lovász. This one is fun and easy to read. You'll get introduced to a myriad of topics in discrete mathematics which are fun and you won't usually study them at your typical courses in college.
Finally, I would like to recommend Topics in the Theory of Numbers by Paul Erdös. This book is rather challenging, but in my opinion gives you a feeling and intuition for Number Theory unlike any other, with beautiful and ingenious proofs and challenging exercises.
Focuse on cramming the first, and enjoy the others. If you spend a considerable time on these books before going to college you'll be ready to start studying the traditional subjects from more advanced textbooks. You will have the required maturity for Baby Rudin and you'll sail through Axler's, then I hope you'll be fine to tackle Spivak's Calculus on Manifolds and enjoy one of the best experiences as a math major.