You don't need it to read .01v so that meter is fine for 12v/0v. As he said first set it to dc voltage. Just to get an idea go stick the leads directly on the battery as this will not only test the meter/battery but allow you to see what the resting voltage of the battery is. It does not matter if you have red to red, black to black if you have them flipped it will only show -12v there will be no sparks or smoke.
From there go back and get behind the switch. Ground the black lead on the meter and then take the red and check all the wires on the switch, with the switch off. One of those wires should show 12v on it, if not then we have a problem before the switch.
If you have 12v then pull the switch and again measure all the wires and one of them that previously showed 0v should now show 12v. If you have 0v out on all the wires then we know it is the switch, if you have 12v out on a wire then we need to move to the multifunction switch.
Seriously don't let it scare you I do not know how you could do any serious damage/harm unless you break something taking it apart.