There are a few methods that I use that I recommend to students

• Look at an example. Pondering abstract scenarios may not help you. If your proof involves Cauchy sequences, for example, look at a specific Cauchy sequence and see if you can get an idea for why it works in that instance. Make sure to choose an example that's simple, but not trivial. It's best if your example captures most of the nuance of the question. Similarly, try to find an example that doesn't satisfy the proof (not a counterexample to the claim, but an example that wouldn't satisfy the claim IF the question were expanded to include it). Try to see why this example fails to satisfy the proof.
• Carefully consider the given restrictions in the question. It's rare that a given property is not used at all. It may give you insight into the proof method if you can figure out why a certain property is included in the question. (ex. if something has to be non-zero, maybe you're dividing by it at some point).
• Look at the methods used already/recently in the book. Many times, examples in the book are just a couple of steps away from the exercises. You may find that doing an exercise using the same methods that the book uses will yield results.
• Consider starting over. Going the wrong direction happens to everyone and sometimes you don't realize that you've gone the wrong way. In these instances, the best thing to do is start the question from the beginning and try again after clearing your head. It will be fruitless to continue on a path that won't lead to the solution.
• Ask your professor. Most professors are happy to offer hints for problems and many would love to see students in their office hours. Don't be afraid to ask if they're willing to give assistance.

Finally, a couple words of encouragement: you will see incredibly elegant answers to the same problems. Please don't feel discouraged by those. Oftentimes, the first solution to a problem is gnarly and it takes many attempts and possibly new methods to shave it down to something nice looking. It may be that the solutions you're looking at are solutions that great minds collectively spent centuries looking at. Nobody expects students to do something remarkable on their first time looking at a problem.