I intend for this post to serve as one place to get ideas (the vast majority of which are quite unoriginal with me). Naturally, no one solution method can possibly work for every problem. I would argue that such a solution method does not exist. However, the ideas that so many people have culled together over time may be of use to you, and so I present a few of them here.
Firstly, the Basic Problem-Solving Strategy is a great way to organize your work. It might even work well for more advanced problems. Naturally, not every step will necessarily apply to every problem. This strategy is modified a bit from the problem-solving strategy found in Young and Freedman's University Physics with Modern Physics.
Secondly, there is the famous George Polya book How to Solve It: A New Aspect of Mathematical Method. This is a classic work that every mathematician, scientist, and engineer should read.
Thirdly, I would also highly recommend Michalewicz and Fogel's How to Solve It: Modern Heuristics. This book attempts to pick up where Polya left off; it has some delightful passages.
Finally, I would recommend looking into the 40 TRIZ Inventive Principles. I think there are some great ideas there. Once you've read the link in the previous sentence, you might also want to check out the TRIZ Matrix, where each principle is matched up with the contradictions it might be able to solve.