Phuket International Airport Arrivals
Phuket airport is not very big, so arrivals are generally simple. Note that if you're flying to Phuket via Bangkok with your luggage checked through, you will go through immigration formalities in Bangkok, but collect your bags and go through customs in Phuket.
Domestic Arrivals to Phuket
Most flights will park at a terminal sky-bridge. On entering the terminal, you'll be directed to baggage claim, which is downstairs. If your plane parks on the tarmac, you will enter the terminal building at baggage claim.
Baggage claim is quite small so it's usually quick and easy.
Phuket International Arrivals
If arriving on an international flight, you first go through immigration, then proceed to baggage claim to get your luggage. After retrieving your bags, pass through customs to reach the public area of the terminal.
In the "high" season from November to February, several international flights can land very close to each other, resulting in long queues at immigration.