I don't know much about troubleshooting but I have the same equipment as you. I have found when troubleshooting things a digital oscilloscope would be very useful. Digital scopes can single shot capture things and that is what you need for troubleshooting. You can still troubleshoot with out one it is a lot more difficult and limited.
I've done an awful lot of troubleshooting over many years,& the number of occasions when it would have been useful to "capture" things is vanishingly small.
I've only done a little electronics troubleshooting over the last year, and I've found being able to capture single-shot activity with a DSO useful when starting to debug digital communication and checking bench power supplies I'm refurbishing for proper power-on behavior, etc. In some cases, I might be able to get things to repeat at regular intervals so I could watch on an analog scope, but that would take added effort I'd rather put into figuring out what was wrong and fixing it.
I've found troubleshooting bench older power designs and HP power supplies to be a good starting point for me.
I've been fixing (and breaking) things my whole life, and I've found the general cycle is to observe the improper behavior, then start putting it in context of the overall system, then use that perspective to look more closely at the behavior, perhaps trying to perturb it, and drill in more deeply to the relevant areas of the system. The Power Designs and HP service manuals I've looked at have a good Theory/Principle of Operation section to help you understand the overall system. Its probably worth printing out the relevant sections (block diagrams, schematics, layouts, etc) so you can spread them all out in front of you to refer to as you read through. Make note of test points and expected values if they aren't already marked.