The immigration office is located in Phuket Town, near Sappan Hin park. They process all applications for visa stay extensions. From the gym it takes about 20 minutes by car. They are open Monday-Friday. The immigration staff take a break from 12:00 - 1:00pm for lunch. Most extension applications require a photo. If you don't have one you can have one taken there.
Most people can come into Thailand without a visa but I recommend to everyone to get a tourist visa BEFORE coming to Thailand. It's free and good for 60 days. With a tourist visa you can get it extended at Phuket immigration for another 30 days, WITHOUT having to leave the country. It's really worth your effort to get a visa beforehand.
