Firstly, I don't think that looking at syllabi is a necessary start.
I suspect that you are getting a math degree because you either think that math is easy or fun (or perhaps both). If I presuppose that you, as an about-to-be-undergrad math student, are exactly where I was when I was about to enter undergraduate math, then I will suppose that you know calculus, trig, and elementary probability really well, you haven't really done proofs in a while or at a sufficient collegiate level, and math is both fun and easy.
Then I think you should do something you find fun. Perhaps Barbeau's Polynomials would be a great start (the reviews say it's an extension of high school math, which is true only in the sense that a high schooler should be able to pick up the book, but I would probably learn still more from it if I were to go through it again even now). It's a problem book. And if you've a good network, you can ask others when you're stumped. Being able to ask others is important.