First of all, anything that can be adjusted by an adjusting screw you can do yourself. If a key gets loose because a pivot screw has backed out, you can certainly tighten that at home. The next step would be inter-key actions that rely on cork (mostly) or felt (some) for timing of key closures. You really need a leak light to get into these. While many professional technicians carefully bend key feet to effect many of these adjustments, the non-technician can sand cork or glue shims to get the same result more slowly.
Then you get into leaks caused by a pad defect. You can't regulate a leaky pad into sealing, you've got to either re-float (which probably won't work) it or replace it.
The hardest part of your question is "how to learn it"? The way I learned this stuff was to observe carefully how the mechanism works - what key does what, which pads close together, how is the adjustment of that timing made, etc. Because every make of saxophone is a bit different, and each individual problem is a bit different, and there are thousands of possible permutations, there's not really any way to put forth a recipe for troubleshooting, as in 1) do this, 2) do this, 3) if A then B; I mean you could write all that down but it would be thousands of pages when the real answer would be "look at the thing and see why it's not working".