Did you change the batteries in the thermostat?
Control voltage to/from (most) thermostats is 24vac. If you happen to check it on dc, you will get zero volts. And, you will often get ambiguous results trying to measure that 24vac to ground...the return often is not connected to ground (directly anyways). Many thermostats do not end up with the 'other' side of the 24vac transformer, so there is no reference there at all.
While there are conventions on how the leads are labeled, it's safest to review the system's schematics. The 24vac transformer must be outputting 24vac (often, it's a bit higher than that) for anything to work.
In a typical thermostat, one side of the 24vac transformer comes in, and when say you turn it to fan, it connects that voltage to the fan circuit, it goes through a relay in the furnace, energizes it with the circuit being completed because the other side of the relay's coil is connected to the other side of the 24vac transformer.
Same idea with the heat, connect that 24vac back to the furnace, goes through (usually) a relay, turns the things on.
Again, same with the a/c.