Ok looks easy enough but on the new thermostat for the red wire it has a jumpered clip from factory. I took a photo but before I start hooking up I'm going to see if YouTube has some videos on installing. From what I've seen wires have been color coded to the thermostat.
R is one side of the 24v control circuit. C is the other (don't cross the streams)
RC = control for cooling C
RH = control for heating H
R to Y is cooling
R to G is fan
R to W is heat
C at stat is usually optional, this provides 24v power instead of batteries or in addition to batteries
Just be sure furnace and thermostat wires connected to the certain terminals match. Typically G is green, Y is yellow, R is red, Common (C) is brown or blue.
If the furnace and the ac BOTH have their own transformers you would use them independently. This is rare, I've only seen this when you have radiant floor heat and a central cooling only system. (Mansions)