If you want to send mail, post offices and parcel services are widely available. In Phuket Town, there is a post office at the corner of Phang Nga Rd and Montri Rd. In Patong, there is one at the appropriately named Soi Post Office, a side-street of Thavee Wong Rd (near Molly Malone's).
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. You can also pre-purchase a Thai SIM card online or purchase one at one of the many phone shops all over Phuket. Pay phones are uncommon, as most Thais have mobile phones. Phuket has very good mobile phone coverage. Mobile Internet is available from all providers. The area code for Phuket is 076. Dial 0 if you're calling from within Thailand.