Perhaps the most important part of A/V system trouble shooting is simply a clear understanding of what your objective really is. The objective is to solve a problem and make things happen --- to turn an inoperable system into an operable system. That's something that, first, requires that you actually understand your system. You have to know how something is supposed to work, in order to make that happen.
It's much easier to troubleshoot a system, after you completely understand how it is setup. If you installed your own rig, chances are pretty good that you can operate it. The chances are also good that you can solve any mystery that arises. My suggestion for learning to troubleshoot, is first learn how to build the system from the ground up.
Almost every road dog tech, I ever worked with, could build and troubleshoot with their eyes closed. It came natural. If you spend most of your life, setting up and breaking down systems, there won't be much you don't know about it. Once you understand signal flow, especially within your own system, life gets much easier.
The guys who were always in panic mode, who couldn't figure out whether or not a channel was muted, were always volunteers who had never set up a rig. They hadn't been there long, they hadn't been trained or taken the initiative to read a manual or ask a question. Most of them were perpetual amateurs. They were behind the board because of their desire to help and serve, but they had no idea what was going on.
More of that "zeal, without knowledge" business.
I had a church call me once, in desperation. We had installed their system about two months earlier and it suddenly wasn't cooperating. The pastor was stressing out. He said nothing is working. I asked a few questions over the phone, and figured out the issue. There were two button on his system that had been changed. His phantom power switch was off, so none of his phantom powered mics were working. He also said that nothing was coming from the main speakers. It was a Mackie board with something called a "break switch." It is used by clubs to shut down everything except the background music. That had been pushed, and muted all other inputs.
He insisted that he checked them both. I insisted that he didn't. He insisted that I drive one hundred miles one way, on a Friday afternoon, to push those two buttons. I told him how ridiculous that service call bill would be, and he didn't care. So, I drove two hours, pushed two buttons and drove two hours back.
We trained every potential operator when we installed that system. But, all the training in the world is no substitute for actually placing your hands on something and doing it. The best church techs were usually the guys who wanted to help with the installations or upgrades. Those folks rarely called for help later.
One day during the first week when I was in recording school, we were ushered into a tech lab. It was a computer-based recording class, built around Mac computers and Yamaha keyboards. When the instructor opened the door for us, we were greeted with a small mountain of gear, piled up in the center of the room. Knowing that we all claimed to have a certain amount of tech skills, before enrolling, he told us our mission. There was enough stuff in that pile, to assemble twelve complete computer workstations, complete with identical racks and keyboards. He told us to get busy and then left to get coffee.
By the end of the day, we were all experts on building computer-based recording systems.
When I built my first system, I didn't have a clue what I was doing. We guessed and assumed and plugged things in randomly. Eventually we made it work. Eventually, I began to understand what was going on. Eventually, after years of doing it, I got pretty good at it. Now, I can walk up to almost any system and get it operating fairly quickly because I understand how it works and what I am trying to accomplish.
Don't miss that part. You can't effectively troubleshoot anything, if you don't understand what it is supposed to do.
The fastest way to learn troubleshooting is to learn to set up a system. The gear is different on every rig, but the principles are the same. Understanding your inputs and outputs is critical. What goes where? Understanding how the system is supposed to function makes life easier. What is it supposed to do?
The most likely problems, in every system failure, are either operator error or cables. Maybe ten percent of service calls were actual problems with the gear. Most of the time cables were damaged or unplugged, or the operator just didn't understand the system.
If I had to train a helper, someone who didn't know anything, I would start by handing them a portable system. I would zero it out and build it with them. Then tear it down and make them do it alone. Nothing is more effective than getting your hands dirty. If you want to learn to repair a system, start by learning to build one.
A solid understanding of signal flow and a basic understanding of what we are trying to accomplish, will usually be all you need to make it happen.