This is for a 120v hot tub. Do I need to use size 6, 8 or 10 gauge wire?

How many amps?

20 amps

#12 should be fine.
How is it heated?

Has elc heat.. all in 1 unit # 12.. was not sure.. so I can us a 20 amp breaker or a 15 amp...

I know that it would be overkill but do it maybe as a 30 amp circuit

12 gauge wire will handle a 20a breaker.

The purpose of the breaker is to protect the wire, by preventing too much amperage from flowing through the wire and causing a fire.

A 30 amp circuit will require a larger gauge wire and would be a complete waste of resources for this application

I have a hot tub that runs 240volts / 40 amps / 60 Hz. I need about 125 feet of wire to the breaker. Should I be using AWG 8/3 with ground wiring?

I would use 2 #6's, 1 #8, and a #10 for ground in 3/4" PVC. Other areas may allow the 8-3 or 6-3.