I second most of what the other answers have said, and would like to add a technique that I think is very useful for people first learning how to prove things:
If you are trying to "prove statement X," take the point of view that you are unsure if statement X is true. Then, try to decide if it is true or not. Seek counterexamples, as Hagen von Eitzen suggested. Seek evidence that might suggest X is true as well. If at some point you become convinced that X is actually false, great! Try to convince somebody else. If, on the other hand, you become convinced that X is true, great! However you became convinced can be the basis of your proof.
The heart of this piece of advice is: you need your proof-writing skills to be linked to the process by which you come to believe what's true and what isn't. Learning how to prove is nothing more than learning how to write down an absolutely convincing argument. Math has developed a lot of techniques, tricks, common argument patterns, etc., giving the impression that there is a whole body of stuff one has to master, but at its heart, a proof is nothing more than a logical argument that serves to convince everybody that something is true. To learn how to make good arguments, you need to be tuned into what is convincing and what isn't, and the authentic way to do this is to stay tuned in to what convinces you and what doesn't. So in trying to create a proof, the best thing is to take the point of view that you aren't sure if it is even true, and actually decide for yourself if and why you think it is, being as skeptical as you possibly can. If you become convinced it is true, no matter how skeptical you try to be, then whatever convinced you can be turned into your proof.
As an aside, I believe that those of us who are experienced at writing proofs have all, at least on some (conscious or unconscious) level, developed this habit of taking the point of view that we are not sure if it is true. Then we write the proof to convince ourselves.