Tanjung Pandan is the main city on the island.
Daily flights connect Tanjung Pandan to Palembang, Jakarta, Pangkal Pinang and Batam. These routes are served by Sriwijaya Air, Batavia Air, and Riau Airlines. The flights from Jakarta both depart around the ungodly hour of 7 AM.
Taxis are non-existant and public transportation on the island is hard to find. Hence the only practical way of moving around is to rent a car or a motorcycle. Expect to pay Rp 400,000 - Rp 600,000 per day (all inclusive) for a good car with driver.
Enterprising (and free) locals drive to the airport at flight times, so you can haggle with them to get you to your hotel and arrange the rest of the transportation from there. Or ask your hotel to pick you up from the airport (which the upscale ones will do anyway).
The flights back to Jakarta leave at around 8:30, so missing them means another day at the island.