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 all over the island for around 200 baht a day (2016). Almost every rental agency will ask for a deposit of your original passport. If you don't feel comfortable handing over your passport to an unnamed shop at the side of a road then give them your driver's licence card instead. Some of the shops will happily accept it thinking it's your passport/ID card. Petrol is available on every street corner for 40 baht/litre.
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.
Otherwise, plenty of tuktuks and songthaews are available on the main road. A typical fare for a ride from Saladan to Klong Dao Beach is 40 baht.
Jeeps can be hired for 1,200 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.