I can't guarantee what I'm telling you is 100% correct but I am hoping to be a paediatrician and this is what my careers teachers and research into universities has told me: Before you specialise in paeditrics you need to study medicine. Most universities ask that prospective medical students must have A level Chemistry, and either A level Mathematics, Biology or Physics and one other A level subject (for most universities this can be whatever you want but a few ask that it is also mathematics, biology or physics). However, for the two of the subjects (mathematics, physics or biology) you don't have at A level you need to have a pretty good grade (usually a B or above) in that subject at GCSE (or if you don't take individual sciences you need a good grade in double award combined sciences). I hope this is of help to you, I was completely overwhelmed by the information I was being given when I started sixth form. Good Luck.
To become a pediatrician you must complete Medical School. Princeton University does not have a Medical School. If you intend to be a pediatrician and want to attend Princeton, you would take a Bachelor's degree as a Pre-Med student and then apply to a Medical School, eventually specializing in Pediatrics.
There are no particular classes in high school that are required to become a pediatrician, so you can take just about whatever you want. The main thing would be to do well enough to get into medical school. If you really wanted to take courses that would make learning what you need to learn in college easier, I would suggest math and science courses. You will have to take a bunch of these in college and they help you to think analytically.