My thermostat will not trigger the ac to start unless I move the thermostat up to a higher number and than back down to a lower temp than it will start the blower. Is there some kind of relay switch or something that is not working? The AC will shut off when it gets to the set temperature but than will not automatically start again

May be the themostat.
It supplies power to outside compressor as well as inside blower.
If you have 24 volts at outside contactor coil, and it doesn't start, it is at the outside unit.