Ganging Two Single-Pole Dimmers
Turn off the branch circuit breaker. Use the non-contact voltage tester to verify that the circuit is safe. The non-contact voltage tester will beep loudly if there is still voltage present in the gang box.
Cut the stripped ends of the insulated wires in the gang box flush with their insulation. Good wire splices start with straight wire ends. In the case of new wiring, go to the next step.
Remove 3/4 of an inch of insulation from the ends of the insulated wires in the gang box. Use a scrap piece of the same size wire to adjust the cutting depth of the wire cutters. You want to set the depth to cut cleanly though the insulation without biting into the wire. "Ringing," cutting a groove around the wire, will weaken the wire and cause it to break. Ringing also reduces the wire's cross-section area increasing its resistance to the flow electric current. You want to avoid damaging the wire when stripping it, so take the time to get the cutting depth right.
Splice the three white, neutral wires together. Hold their stripped ends side by side and twist them together with a right-hand twist using the Lineman's pliers. Complete the splice by screwing on a wire nut.
Splice the green leads from the dimmer switches to the three bare copper wires in the gang box. Make this splice the same way you spliced the white, neutral wires together in the above step.
Splice the 'Line" side leads of the dimmer switches to the black wire coming from the service panel. Splice the "Load" side lead from each dimmer switch to the black wire going to the light it will be controlling. Make sure the you do not reverse the dimmer leads marked "Line" and "Load." Reversing the leads can damage some electronic dimmer switches.
Secure the switches in the gang box and install the faceplate. Turn on the branch circuit breaker.
Ganging Two 3-way Dimmer Switches
Follow Steps 1 through 6 from "Ganging Two Single-Pole Dimmers".
Splice the red and black "Load" leads to the "Traveler" wires continuing to the three-way or four-way switch. The circuit traveler wires are not polarized. It does not matter which Load lead you connect to which of the circuit wires as long as they are coming from the same cable. Do not mix the travelers from one cable with those coming from the other cable. No damage will be done if you do, but the switches will not operate properly.
Repeat Step 7 from "Ganging Two Single-Pole Dimmers".