Often the most difficult part of solving a math problem is doing the very first step. After doing many problems they begin to become routine because there is only a finite number of techniques to be utilized. I will attempt to classify as many of these techniques as possible. The techniques will be sorted by category. Look at special cases.
Numerical data is great stuff.
If you are supposed to find and prove a result, try to find the answer first, then prove it. When trying to prove something is impossible, look at it mod something. Bases are how we build numbers. Where there's a P there must be a Q. If you are given a polynomial, look for one closely related to it which is easier to work with. Try assuming the result is false and derive a contradiction.