Is there a correct order to learning maths properly? I am a high school student but I would like to self-learn higher level maths so is there a correct order to do that?
I have learnt pre-calculus, calculus, algebra, series and sequences, combinatorics, complex numbers, polynomials and geometry all at high school level.
Where should I go from here? Some people recommended that I learn how to prove things properly, is that a good idea? What textbooks do you recommend?

Quite often the transition to higher, pure math is real analysis. Here proofs really become relevant. I would suggest this free set of down-loadable notes from a class given at Berkeley by Fields medal winner (math analog of Noble Prize) Vaughan Jones.
https://sites.google.com/site/math104sp2011/lecture-notes
They are virtually verbatim and complete as a text. They build gradually so you can get a good base. The material is Prof. Jones's own treatment and the proofs are quite accessible and beautiful.
You might just give it a try and see if it works for you.
