I have an old natural gas heater that works very well, and I expected it to be just be two wires for my thermostat. A pretty straightforward process. And I don't think it's too complicated, but there was a third wire in there.

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I'm assuming the connection between RC and RH is just a jumper to get them to act together.

My question is, if I get a new thermostat with an R, RC and an RH, should I just wire straight to the R, and forego the RC/RH, or should I do the same setup of wiring into the RH and jumping to the RC, ignoring the R?

Almost all of the home wiring I've done in my life has been for the basic outlets, light switches, and circuit breakers. Stuff that is pretty standard. This doesn't seem to be too far out of that realm, and it's not something I'm SUPER worried about. I just don't want to blow the transformer if I can help it.

Edit: Hooked it up, and had no problems, as expected! Thanks everyone for their response.