The most direct way to get to Koh Lanta is to fly into Krabi Airport and transfer overland to Koh Lanta. Although "overland", the trip will still involve two short ferry crossings. You can plan on it taking around two hours, depending on where on Lanta Yai you're staying and how long you have to wait for the ferries. A taxi (private car) transfer will set you back about 2,300 Baht (59.73 USD), while a van (shared taxi) will cost about 800 Baht (20.78 USD). Check with your hotel, since some of the more luxe resorts may include free transfers with the room.
You can also reach Koh Lanta by boat from Krabi town or Phi Phi Don Island. The boat isn't really any faster, and in the rainy season from June to September the seas can be quite rough.
Ferries from Ao Nang, Krabi Town, Phi Phi and Phuket (via Phi Phi) all dock at the pier in Saladan, on the northern tip of Lanta. You will need to take a taxi from here to your hotel, and there are always plenty of them on hand waiting for each boat.