How do I make a 110V fan turn on when the thermostat turns the AC on?
All you'd need is a relay.  You'll need one with a coil rated for 24VAC (or whatever the thermostat voltage is), and a contact rated to handle the current draw of the fan at line voltage.  


Connect the ungrounded (hot) feed to the Common terminal of the relay. 
Connect the ungrounded (hot) wire for the fan, to the Normally Open (NO) terminal of the relay.  
Connect the grounded (neutral) from the feed, to the grounded (neutral) of the fan.
Connect all grounding conductors. 
Connect a wire from the G terminal on the thermostat, to one of the coil contacts on the relay.
Connect a wire from the C (common) terminal of the A/C transformer. 


When the thermostat calls for COOL, it should turn on the fan as well.


Didn't add it to the diagram, but it's probably a good idea to have a fuse on the R wire.