I recently moved into a house that has a gas furnace. I noticed when I moved in that the a/c wasn't blowing well, or very cold. When the gas was turned on a few days later, the a/c vents started blowing harder & the air got colder. Is this because the furnace is controlling the a/c? I have a switch to turn the furnace off, and when it goes off the a/c still runs... I don't know anything about heat & air, how does this all work?

The fan should operate the same -- when the thermostat calls for heat the fan runs when

Needed -- when the thermostat calls for cool the fan runs -- should be the same

By the a/c you mean the outside unit?

Typically the disconnect located by the furnace will prevent both the outside and inside units from running, but the outside unit still has power. That unit is controlled by a separate breaker.

You could have a dual fuel setup, although rare. This consists of a heat pump outside with gas backup.

Another possibility is that it's a dual transformer system.

The thermostat on the wall somewhere controls the mode of the furnace and whether the heat or a/c comes on. The fan can be on all the time if desired.