As for learning resources, I came across a decent article on the subject when I was starting out (1), and most of the oscilloscope manufacturers have whitepapers on SMPS diagnostics, the Tektronics one I read a while back (2) gave a good overview. A lot of the whitepapers have a manufacturer-specific focus, but they still have good information that can be applied to almost any oscilloscope.
If you want to get really into the power supply and do high-side measurements you'll need an isolated differential probe, which can cost as much as an inexpensive oscilloscope, but for DC output measurement you shouldn't need anything special. Current probes are a lot more affordable if you're interested in looking at loads or current fluctuations/harmonics, but that's more useful after you've figured out a bit more what specific properties you're trying to measure.
1: https://www.testandmeasurementtips.com/test-switching-power-...
2: https://download.tek.com/document/3GW_23612_7.pdf
Edit: I forgot to mention that the EEVblog forums are a good resource also, but they sometimes aren't as friendly as they could be towards people just starting out.
Any recommendations for learning resources that could help with understanding DC power supply analysis for non-EE types? While refurbishing laptops and working with microcontrollers I’ve run into some odd things where ruling out transient power supply issues would probably be helpful.