Mowgli your drawing is pretty close. In order to trigger the relay the switch has to have its own power source. The switch then sends this current to the relay when its "flipped", causing the relay to close, therefore completing the circuit allowing the current from the battery to travel to the lights. And because relays hardly draw any power (typically < 1amp) you could probably get away with using 18awg wire or smaller, but to be safe I would stick with 16awg for all wires running to and from the switch. For everything else use 12 or 14awg.
Also, since your switch is three pronged you don't need to worry about independently wiring the light for it, it will light up automatically when you turn it on. The downside to this type of switch is that it won't light up while its in the off position.
If I were you I would tap into the HVAC or cigarette lighter fuse to power the switch. That way you can only turn the lights on while the engine is on.
