The minimum wire size for my hot tub with a 40 amp GFCI is #8 gauge/4 wire, but I will have to have about 125' of wire to go from my breaker box to the hot tub. Is the rule of thumb for a wire 100' or more, to increase the size of wire to # 6 and a 50 amp GFCI ? And can I assume that I cannot wire the hot tub and plug into a 30 amp # 10 wire outlet that already exists since it is only rated for 30 amp?

If the tub draws more than 26 amps or 6240 watts, a 40 amp circuit is needed.

Assuming the max steady load of 32 amps or 7680 watts for a 40 amp circuit, the #8 will produce 2.5% voltage drop, the max recommended is 3%.

If your supply voltage does sag any, this percent will increase. Plus the motor will have starting current to consider.

Upsizing to #6 for this length would be good. Still only need a 40 amp circuit breaker thou.

You can start the circuit at the panel with a standard breaker, and have the GFI at the tub, will prevent nuisance tripping due to circuit length and convenience of having the reset close to the tub.