Thermostat is completely blank. How to troubleshoot
Since you have a multi-meter, you can check the voltage of the wires at the thermostat.  Make sure the meter is set to AC (Alternating Current) in this case, and if there are ranges, something safe for 24v.  Probe the Red and Blue wires and see if there is 24v.  You can also check the voltage between the red and any other wire.

If there is no voltage, the low-voltage power supply for the furnace is bad or not getting any power.  There is probably a breaker in the main cabinet for the furnace, but there will probably be a breaker located on the furnace/air handler as well.  Make sure they are both on and recheck the voltage.

The power supply will typically look like this and should have two 120v wires and two 24v wires coming off of it.



If you can find it, check the voltage going to it to see if 120v is present.  Troubleshooting will move on from here if there is not 120v.

Thermostat information

Assuming the wire colors you have are standard...

To turn on the fan, the thermostat makes a connection between the red and green wires.  If you place a jumper between these wires, the fan should run.

To turn on the heat a connection is made between the red and white wires.  The call for heat will always be accompanied by the call for fan.

The blue wire doesn't signal anything - it just supplies voltage to the thermostat in conjunction with the red wire.  To draw a parallel with your house wiring, the red wire is like the black "hot" wire, and blue is like the white "common" wire.