I think the thing to understand is that there are common patterns to proving things that are almost independent of what you are trying to prove. Once you have i

I think the thing to understand is that there are common patterns to proving things that are almost independent of what you are trying to prove. Once you have internalized those it gives you a suite of approaches to proving things. For example, you should try to understand what is meant by a direct proof, a proof by contradiction, a proof by induction, a proof by contrapositive, etc. If you can find a good book that discusses theses kinds of patterns and gives practice questions for employing them with fairly basic mathematical concepts it can really help (and if nothing else will really improve the speed with which you understand other proofs).