Flights from the west coast are currently (as of June 11th) running $590(+tax) to Bangkok and $690(+tax) to Phuket international. The better travel discount shops have assured me that these prices will drop by around $70 towards the end of the month. The shops currently listed are not necessarily the best choices. I am still looking for a better fare.  I have not yet bought my tickets and would suggest holding off a little longer.

Silver Wings Travel
Airline -
Bangkok -
Phuket -

Bi-Coastal Travel - prices checked 7/11
Airline China Airlines
Bangkok 611 + tax (approx $60)
Phuket Above + 100
 

Sonic Travel - Prices Checked 6/12
Airline - Japan Airline or Thai Airways
Bangkok - 590
Phuket - 690 via Bangkok
Note: Depending on the airline the tax varies.  Even though Thai Airways is cheaper, I believe you do not get to get off the plane in Japan, Something that is definitely worth the $10 extra.

Tax Thai Air 58
Tax Japan Air 68

Cathay Pacific 810 direct Phuket via Taipei.

Rock Bottom Fares - Prices Checked 6/12
Airline Japan Airlines
Bangkok - 820
Phuket - 945 via Bangkok
Note: Their name does not seem to be reflected in their prices.
 

New Global Travel - Prices Checked 6/12
Airline - Korean Air
Bangkok - 620 + Tax 52
Phuket - 910 + Tax direct via Seoul
 

JustFares - Prices Checked
Airline
Bangkok
Phuket