Fault-finding is actually such an extensive topic, it's quite hard to write a summary guide in a forum post. If I had to single out half a dozen bullet points:
* Understand how things are supposed to be when they are working. E.g. If you don't know what reading something is supposed to give under normal conditions, you won't know if the reading you are getting now shows a fault or not. If necessary, work this out first, away from the installation, so that you have a basis for comparison before your (possibly odd) findings start to cloud the picture.
* Pay attention to every available source of information, even even if they conflict. Listen to anything anybody tells you, but don't trust any of it initially. Look at the installation from every angle, touch it, smell it, poke it with testers and make good note of the readings even if they don't seem very interesting in their own right.
* Be critical of your own findings. It's no good saying to yourself that there absolutely must be continuity because you tested it five minutes ago, if you happened to have put your test leads on the wrong terminals. Accept that some of your tests and deductions will be wrong, do not hesitate to revise them to the point of complete U-turns if needed.
* As you work, allow multiple theories to exist in parallel, rather than forcing yourself to believe only one possible explanation that you favour. If you are too focused on proving that one theory right, perhaps because it was the right answer last time, you are more likely to miss clues and opportunities to follow more promising leads this time. That said, don't ignore clear evidence that it's the same thing again, leading to...
* Occam's Razor: The simplest valid explanation is often correct. Periodically stop investigating and review what you have discovered so far, juggling the pieces of the puzzle to check whether a simple alignment of a few of them produces a likely solution.
There is no substitute for experience and knowledge, it takes years to get good at this, so don't worry if you find yourself turning down blind alleys at this stage. I have the advantage of getting a lot of practice in a very wide variety of situations. Rarely a day goes by that I am not called to investigate something, and despite doing that week-in, week-out for 30+ years, I'm still learning new techniques and tricks and shortcuts.