household t-stats run on 24vac. i made a "reptile room temp controller" with a Honeywell stat, transformer, several relays, and wire.
I use a auto changeover stat that will kick on heat or ac(in the future) depending on the temperature of the room. yes temp swing is high around 3 degrees, I use a data logger to ensure room temps and make sure no faults are found. also has a fan switch as well as a hacked habba mist to run a 120v pump for misting.
for heating I use a oil room heater with fan blowing through it.
a/c nothing yet... since its winter
I recommend being careful on trying to do this yourself. since you will be working with line voltage. you also have to keep in mind the watt's being used by heaters and etc. if you undersized a relay it will burn out quickly or melt down.. plus most residential outlets are rated 15 amps. Which oil room heaters max out(1500 watt).. but luckly setable to medium(800 watt) and low(400 watt)..
As for incubator control I would recommend the herpstat.