[kraitis]

You want to acquire and shoot for the so-called mathematical maturity. More precisely: to become an autonomous problem-solver and have the know-how to solve (non-)trivial proofs. Typically this means bridging the gap between computationally based maths which one is exposed to in pre-school to high-school years and sometimes in the first year of college/uni, and proof-based maths which involves and demands a good command of sets and operations on sets, quantifiers (universal, existential), logical operators (not, and, or, material conditional, biconditional), and proof methods (direct, indirect a.k.a reductio ad absurdum, induction, pigeonhole principle, etc.)

A good series of books aimed for pre-school and high-school students to accomplish just that is The Art of Problem Solving. Google it.