First eyeball the rear derailleur to see if it looks clearly misaligned, but otherwise I'll shift through the RD to see if any correction is needed or if it seems like an obvious, easy fix (like cable tension) and check limits and b tension. If it's being troublesome at all, I remove the derailleur, align the hanger, and reinstall the derailleur. This is a good time to just move the derailleur with your hands and see if any pivot points have either excessive play or turn roughly--potentially lubricating pivot points can be helpful. Eye rear derailleur for alignment (it may be bent even when the hanger is straight)--ideally replace if possible, otherwise try to adjust with correctly aligned hanger, and maybe if really in a pinch try to bend derailleur/hanger so that hanger operates in alignment. Then I loosen the cable anchor and assess cable.housing friction as well as I can. Replacement of cable and housing may be advisable. If replacement isn't strictly necessary or approved by customer and if it's easy to do so, I'll use this as an opportunity to lubricate the cables. Then adjust the HL with the cable uninstalled (center of guide pulley to outside edge of cog). Set barrel adjustors in and pull cable reasonably taught. Shift into the next easiest gear and roughly align guide pully under that cog. Shift into biggest cog and adjust low limit so that limit screw is as far in as possible while not adversely affecting shifting. Adjust b screw--typically very close to cog on most 2x or 3x setups, potentially farther out for wide range cassettes (15mm for SRAM 1x 11-12sp, for example). Shift through all gears and fine adjust cable tension so that shifting is balanced to be as fast as possible for all shifts.

For front shifting, again, assess performance first. Before getting too deep--look at the derailleur and check for excessive play in the pivot points and seriously bent looking cage and advise replacement if necessary. If it's not working well, first check to see if the derailleur is as close to the chainrings as is possible--ideally 1-3mm above the large ring. Sometimes on 3x systems you have to check to make sure the inner cage clears the middle ring when in the big. For most systems (SRAM Yaw is an exception) the outside plate of the cage should be parallel to the chainring. Loosen cable and set derailleur into ideal position, reattach cable. Set the inside front limit so that it just barely doesn't rub in the small chainring and big cog. Shift into the middle chainring and adjust cable tension so that there is minimal rubbing at both extremes of the cassette. Shift into big chainring and adjust limit so that it barely doesn't rub when in the small ring--increase cable tension if necessary to get it into this position. In an ideal world everything should now shift great--you may have to slightly loosen the FD limits to get reliable shifting, and you may have to alter the cable tension a bit to get the best possible shifting. You may need to adjust the angle of the FD slightly to improve shifts. To judge this, look at where the chain needs additional contact to make the shift.

If you want to help people out with this kinda garage wrenching, probably would make sense to get bulk der cables/housing. The Park der hanger tool is sanely priced. Unless you have a pile of old parts, learn to recognize when to kick it to a coop or a lbs when they really just need some new parts.

As an example of calling it, I work professionally as a wrench and had a customer come in with some 10sp Shimano drivetrain. Front shifting was atrocious. Dorked around with setup for a while but called it pretty fast when I determined it was in a pretty good adjustment and that the cables and shifter was fine. Grabbed a hold of the derailleur and found the pivots were really worn and loose. New der and shifting was amazing right off the bat with a basic, sane to spec adjustment.

So don't beat your head against something that just won't work even though it's set up seemingly correctly.