the elegant universe is a pretty good book to introduce yourself to modern physics and string theory
don't worry if you don't entirely understand what he's trying to explain, even the concepts. Just worry aobut the background information about the direction study is headed into
as for math it's really hard to know what to look for as higher level maths can really head into many different directions
Euclids stuff as suggested before is good for geometry and basic introductions to proofs but it doesn't introduce anything really interesting
since my studies in math have been more university guided and I haven't had to rely on looking up ressources I can't be sure of the following but I think looking into Disquisitiones Arithmeticae (Latin, Arithmetical Investigations), would be worth your while.
But it's probably filled with useless stuff you don't want to worry about so what I'd suggest is first looking into different proof methods, the most important being induction
then learning some elementary set theory
then either choosing calculus (which I dislike)
or discrete math (which I like) and then you can look into modulus arithemetic or combinatoric probelsm which are really fun to do if you're an amateur mathematician and not going to suffer through formal training