i doubt you could check anything with the meter because the 20 pin port usually needs to be plugged into the motherboard for the power supply to turn on. first i think you should make sure the psu is set to the correct voltage. most likely that would be 115V. if your psu (at back of ur tower) does not have the voltage setting then no worry about that. make sure the plug is in correctly. you could use your multimeter and make sure the wall outlet has the correct voltage. if you dont know how to do that, dont kill yourself trying, just carry the pc back to where it was b4 u moved it and try turning it on.
check the pins on the board that go to the front power button and make sure they are in correctly.
u could also sniff around inside the system for any burnt components.