Checking voltmeter on my Honeywell Wifi Programmable Thermostat
Are they suggesting that I touch the red cable to the R or G wire screw and them COM into the other wire screw?


Yes.

Voltages are potential difference, they are relative not absolute. You can often measure with respect to protective ground but it is usually more appropriate to measure between two current-conductors (e.g. between circuit ground/common/neutral/negative and circuit supply/line/live/positive)

In extra low voltage DC systems (&lt;50V), screws in the case may not be electrically connected to the circuit under test and so may not be useful in measuring voltages.


  why does my voltmeter sometimes read 13.0 (for instance) and at other times read -13.0?


Because sometimes you connect the voltmeter leads the other way round.