You can usually pick up a free Thai SIM card at the airport in the baggage claim area. Look around for a booth or a kiosk.
The area code for Phuket is 076. Dial the 0 if you're calling from within Thailand.
Pay phones are uncommon, as most Thais have mobile phones. Phuket has very good mobile phone coverage, albeit over slower 2G/2.5G GSM networks. Pay-as-you-go SIM cards can be purchased for a few hundred baht, and local call charges range from 1 to 3 baht per minute depending on the package. You can also pre-purchase a Thai SIM card online.
Mobile Internet is available from all providers, with True Move offering 3G access from a limited network of base stations on the east coast, including Patong, Kata, and Karon. All other networks offer EDGE and/or GPRS access, so don't expect fast Internet connections on your mobile device. For email and basic surfing GPRS speeds are normally fine, but access to websites hosted outside Thailand can be slow.
Unlike many Western countries, Thai networks bill mobile Internet usage by duration rather than bandwidth, with PAYG users paying around 1 baht per minute. Most networks have PAYG monthly unlimited GPRS deals for around 750 baht, and you can easily purchase a SIM and the Internet package at one of the numerous phone shops around Phuket.