There is no public transport on Koh Lanta. The island has one main road running north to south along the west coast, and getting around requires renting a motorbike or car. Motorbikes are available for 200–300 baht per day from countless rental stalls along the main beach road — this is by far the most practical option. The road is paved the full length of the island, though some tracks into the interior are rough.
Taxis exist in the form of pick-up trucks that loiter near the ferry pier and at popular beaches, but they are expensive relative to renting your own transport. Songthaews (shared pick-up trucks) do run informally, but you won't find them on a schedule.
The island is about 30 km from north to south, so the drive from the ferry pier at the northern end to the national park in the south takes around 45 minutes on a motorbike. Fuel is available at several roadside stalls and a small petrol station near Long Beach.