What is the most important part in solving a math problem?
I think this depends on the kind of problem being solved, so I will address word problems in particular, with some additional comments on math problems in general. It might be fair to say that everything is important in solving math problems!
If a word problem is being solved, the most important aspect is creating an equation with the numbers and concepts in expressed in the problem. Words in the problem must represent a mathematical operation or the problem cannot be solved. For example, if one is told that a number is three times another number, the word "times" tells us that a muliplication is involved. Another example is the word "less," when one number is two less than another number. The word "less" tells us there is a subtraction involved. As people practice with this kind of problem, the "translation" from words to operations gets much easier.
Assuming that an equation already exists, it is my opinion that the most important part of solving a math problem is writing down every single step. Many people find it easy to do the steps in their minds, but if something goes wrong along the way, there is no way to figure out where it has gone wrong unless each step is written down. I remember that when I took calculus in college, we were given homework problems that involved pages and pages of calculations. If I did not write every single step, I cost myself hours of work.
Another aspect of doing math that is quite important is checking your answer. All equations can be proved by simply plugging the answer(s) into the equation for the variable(s) being sought. Quite often I find that students have used the correct method and have made some trivial arithmetic error that could have been caught by this method.