Some of the islands are so small that you can walk around them. On others, long-tail boats will ferry you for a fee to nearby beaches or reefs, while on Phuket sŏrng·tăa·ou (minibuses) connect Phuket Town with all the beaches. On the bigger islands, there are taxis or túk-túks ((pronounced đúk đúk), but they're expensive.
In many places you'll need to hire a bicycle, or more likely a motorbike to get around.
Be aware that Thailand's roads are some of the most deadly in the world, with over 25,000 people a year dying in traffic accidents, and that motorbike crashes are the number one cause of injury and death for foreign visitors. Island roads are often in poor condition or, as on Phuket, very busy. Drive slowly, wear a helmet (as a foreigner you're a prime target to get pulled over and fined by the police if you don't) and make sure you have travel insurance that covers medical expenses.