I am trying to hook up a 24 volt transformer to a relay does the common or the 24 volt go to the relay

You need both for the coil on the relay to work. How you get it to work and why is up to you. Good luck.

What does this relay opperate and what activates the relay.

The relay powers the thermostat which kicks on the blower motor

The common will go to one side of the relay and the 24 volt will go to the other side after it goes through the thermostat.

One of the lines goes to the relay the other goes to the wires power the thermostat the is no schematics with the thermostat and the transformer schematics just tell which is hot which is common