Most people spend one day of their holiday in Ko Lanta driving around the island on a rented motorbike or 4 wheel drive car (4WD is essential as the roads are very steep and rocky towards the south west of the island).
Once you have your vehicle, you can just take a map and go. Being an island, it's fairly hard to get lost - and if you do, this in itself will be an experience! Inland, you'll find lots of small villages that seem completely undisturbed by tourists, beaches used only by fishing boats, lush landscapes of rice paddies and plantations, as well as a great viewpoint on the way to the fascinating Old Town in the south east.
There is also a network of caves at the centre of the island, just past Klong Nin village, going inland. Called Mai Kaew, these caverns are only accessible with a guide from the family who live near the entrance (signposted). The caves are spectacular, but not for the claustrophobic. You will also need a healthy sense of adventure - some of the wooden ladders are fairly precarious and rickety.