1. Is there always one correct way to solve a mathematical problem?
No, often even the simplest of problems can be solved in more than one way. So long as your mathematics and reasoning are sound and you reach a correct answer or sensible conclusion, your solution will be acceptable. You must make sure you are answering the question and meeting the requirements of the problem.
2. How do I know what strategy to use for helping to solve a problem?
This will vary according to the nature of the problem. Some suggestions:
If the problem is geometrical or trigonometrical a diagram will often help.
If the problem is algebraic the first step often is to form an equation.
If the problem is about patterns or sequences a graph or table is useful.
If the problem is about statistics a table is useful for organising data and then a graph for displaying the data.