I want to start out by saying that I am new at proof based mathematics. I am used to seeing patterns and using them to solve similar problems. However, I have found this is not a very good way to study for courses in abstract mathematics! I have noticed that similar themes in mathematics keep coming up over and over again(for example, homomorphisims of objects). And partially because of this, we also have recurring proof methods. Many times I initially struggle with proofs because I don't think, "What are the proof methods available to solve this sort of problem?".
I have found that stopping to think about proof strategies is the best way to solve a proof based problem, but many times, because of inexperience I lack knowledge about what tools are available to me in this regard. For example, to prove $A=B$, a way to attack this problem is to try to show that $A\leq B$, and also that $A\geq B$. This proof strategy came up today when I was trying to prove $G_{b}=gG_{a}g^{-1}$ where $G_{x}$ is the stabilizer of the element $x$ under appropriate conditions. Anyway, I showed that $G_{b}\subseteq gG_{a}g^{-1}$ and also that $gG_{a}g^{-1}\subseteq G_{b}$. Once I realized how to attack this problem, the problem sort of solved itself. I have noticed that many problems sort of tell you how to solve them.
What are your favorite proof strategies, or ones that you have found extremely useful?