I have the honeywell programmable thermostat, the instructions call for a connection into screws. My old mercury thermostat don't have a calling for this or these wires, so what I need to know is this, which wire (if any) goes in the rh and/or rc connections? If they don't require a connection, what do I do with them? Thanks ahead of time for the assistance!

Connect Rc to Rh and to the wire that went to R of the mercury stat.

Rh and Rc are for when you have separate or dedicated heat or cooling power sources. Typically they are jumpered from the factory.

So if you don't have a jumper, just add one and then tie your R onto either Rh or Rc.