Especially in maths, all Cambridge will care about is maths. In my interview (Trinity) all I was asked about was the maths problems in the pre-interview test. In fact, they didn't even ask about the books I read. With your GCSE grades, they are less important than your A level predicted grades and AS results. It would be helpful to have A/A* in maths and science, but your results in your other subjects don't really matter. If you impress them enough with your maths ability, they won't reject you because you have a C in GCSE English, for example.
As for what to do, it would be too early to start looking at STEP (ideally, you should start it in about a year. I started September/October of Year 13 and that was OK, but I wouldn't leave it any later). As for now, does your school offer further maths? If not, you ideally would want to self-teach it, and some modules (D1 for example) lead on directly from GCSE.
As for recreational maths books to read, I'd recommend Fermat's Last Theorem by Simon Singh (which almost everyone will have read), and The Music of the Primes and Finding Moonshine, both by Marcus du Sautoy.