How do you trigger the HVAC with wiring only?
A thermostat basically just energizes the individual circuits (G (fan), W (heat), Y(cool) ) which in turn (via the furnace controller) powers a relay which provides line voltage to the actual units (like the fan or A/C.  In the case of heat, it instructs your furnace to open the valve supplying gas).

R (or Rh  and Rc) provides the 24VAC power.  To turn on the fan, you'd connect R and G.  To turn on the fan and AC, you'd connect R and Y (or R, G, and Y on systems where the thermostat controls the fan).  To turn on the fan and heat (as you asked), you'd connect R and W together (or R, G, and W if you have say an electric furnace where the thermostat controls the fan).

If you make a mistake you could potentially fry your furnace controller.

As for finding out what is running - yes there should be 24VAC on the circuit if they are turned on.  However note that this is just instructing the logic board in your furnace to turn on the other components; your furnaces controller might not always obey this. For example, some units will keep the fan running for a few minutes after the A/C or heat is turned off (sometimes this is implemented in the thermostat too).

One other thing I will add based on your comment is about the C line.  This is essentially the neutral back to the transformer (or furnace), so you can use it to power your device (most thermostats use this these days).  You should have 24VAC between R and C.

Never connect R and C, this will either blow the fuse on your board or fry the board all together.