I don't claim to be an electrician, but I always use 12 ga for 110, and 10 ga for 220.....I thought 10 ga was required for 220.....
It's really the current, not the voltage, that determines how thick of a wire you need.
In fact, a significant reason electrical transmission lines use 110,000V+ is that by keeping voltage extremely high, they can deliver the same amount of power with much lower current, and use thinner conductors without incurring significant power losses/heating.
Anyhow, what I do for these things is as follows. It's conservative but better to buy a little better than you need than need a little better than you bought!
1. Calculate the maximum current on the circuit by summing the currents of anything that is on that circuit.
2. Add 25% as a safety factor. This is the minimum that I would use for a circuit breaker on that circuit.
3. For the actual wire gauge, I'd take the circuit breaker rating, again add another 25%, and then look at the national electric code
to find what conductor size I should use.
For example, if you have a 15 amp draw, I would put it on a circuit rated for at least 15 * 1.25 = 18.75 amps, so I'd go with a 20 amp breaker. With a 20 amp breaker, I'd look for a conductor that could carry at least 25 amps, which would be 12 or lower gauge.