Internet cafes are plentiful and typically also offer international calls, fax services, and flight confirmation. The connection and speed is generally good. Expect to pay 1 baht/min for Internet in central locations. One baht per minute is typical for predominantly tourist-oriented shops, many of which also offer lower rates for pre-paid blocks of time. In travel agency Tan Tour (50 m to west of 7-Eleven next to the pier) the friendly owner Tomas is famous for not taking charging customers very seriously, so if you stay only a short time you can usually use the Internet for free, or for a longer time you will usually end up paying only 20 baht/hr instead of 40-60 baht.
Next door to the 7-Eleven in Thong Sala at the pier there is free (open) Wi-Fi for everyone in Sweet Cafe, so if you don't mind sitting in the sun you can use it for free. Also in the food court in the Thong Sala night market (actually open all day) there is free Wi-Fi for everyone. It's not difficult to find well-equipped, quiet, air-con Internet cafés that charge 60 baht/hr. Shops that can accommodate users who want to hook up their own laptops can easily be found. Printing (black/white) is usually 10 baht/page (30 baht/page for colour).
When you venture away from the more developed beaches, expect to pay up to 3 baht a minute. It can be cheaper just for staying in touch (Skype/Facebook) to buy a SIM card with a 1GB data package for 1 month for 214 baht (AIS/DTAC), which is enough for mobile Internet, but be aware some beaches do not have a 3G signal (Haad Yuan/Haad Tien, for example).
Mobile phone/SIM cards can be bought and topped-up all around island in many 7-Elevens. Mobile signal strength for DTAC (Happy) or AIS (1-2-Call) is good all around the island. Avoid using the TrueMove network which has very bad coverage.
Some places try to sell overpriced "internet" SIM cards for prices like 100 baht for the SIM card + 230 baht for the data. It's a rip off and there's no need to waste money for those; normal SIM card costs 50 baht, and e.g. 600 MB/7 days data package (dtac) is 79 baht. Refer to Prepaid Data SIM Card Wiki - Thailand for the details about the packages available and current pricing.
Overseas calls can be made from many agencies and Internet shops, as well as guesthouses/hotels and the like. Most advertise a rate of 15 baht/minute (or 25 baht/minute to mobile phones). Pretty much every Internet place will have headsets for Skype use, which will be free if you don't have to call to a telephone.