The main roads on the island are well paved. However the road just north of Mu Ko National Park is still unpaved dirt. That road ends at the park and so it isn't possible to drive from that point up the eastern side of the island.
It's worth knowing the names of the resorts near yours: both the pickup trucks and motorcycle taxis will often want to pick you up or drop you at a nearby resort with a better road.
Resorts and some of the major tourist services like the dive shops own large pickup trucks and transport tourists around in the tray. Very occasionally, they own minibuses instead. If you come in by ferry your resort will almost certainly meet you at pier with their truck; if you're travelling with very small children who you don't want to cling to while you bump along sitting on the side of the truck you might want to warn them in advance that you'll want to ride in the cab: it's generally assumed to be the private domain of the driver.
You will find the locals almost exclusively ride motorbikes; you can hire these in several places for a few hundred baht a day. You can also hire motorcycle taxis with sidecars. These will happily take you on the unpaved roads but the ride can be as slow as walking. A typical fare for a ride from Saladan to Klong Dao Beach is 40 baht.
Jeeps can be hired for 1200 baht per day. Be sure to check insurance, contract terms and the condition of the vehicle. The "roads" of Ko Lanta combined with the driver operation characteristics of some renters are a serious challenge for any car. The frequently available Suzuki Sporty seems primitive and therefore sturdy, and a replacement should cost around 150000 baht.