When my thermostat is set for cool, the ac and furnace both come on and run at the same time.

What kind of furnace do you have, gas, electric, oil,, brand and model number might help also
Does furnace work OK when set to heat?, we need more info

It's a Rheem, model unknown... installed in the house when built in 2002, it's a gas furnace.

Have you done anything to the thermostat lately? Just wondering