To earn AA miles, you can take AA metal to NRT and a codeshare AA flight (JALways) into BKK. You then fly TG from BKK-HKT. Going this route means an overnight in BKK. TG tickets can be purchased on Thai's websites. Domestic tickets within Thailand are the same price everywhere.

A better but more expensive option would be to fly AA to LAX or SFO and connect to Cathay Pacific to HKG and then to BKK. The advantage of this route is that you arrive into Bangkok at about 11 a.m. so that you can connect to the 1:00 pm to Phuket, arriving at 2:20. No overnight in Bangkok required. If you go this route just make sure your ticket is in H class (economy) or better to earn AA miles. I am not sure if this type of ticket allows a stopover in Hong Kong or not.

Good luck and have fun.