I am about to replace my two thermostats. There's a lot of switches on the "control panel" (litterally) for my heating system and I am not familiar with procedures and afraid to turn something off that won't come on again properly. Maybe I only need to turn some of it off? See picture below.
This isn't it. These are cutouts for individual features within your heating system. This will not de-energize the heating system.
You do that over in the circuit breaker panel. One breaker (per furnace?) will shut off the 24V transformer that powers your thermostat.
24V is not hazardous, and if you did manage to short 2 wires together, the worst thing it will do is make a call for heat, A/C, fan, etc. One exception: do not short R and C, unless you like baby arc flash, foul smoke and replacing transformers.
My advice would be to remove R, Rh and/or Rc first, cover it/them with electrical tape to insulate it, then remove C, cover it also with electrical tape, then handle the rest naked. I wouldn't even bother turning off the furnace.