There are no trains on the island of Langkawi.
Car and bus
The airport is situated in the southwest of the island at 13 kilometers from the capital Kuah. From Kuah the airport can be reached by rental car in half an hour via the Jalan Padang Matsirat road (road number 112).
There's no public bus transportation on Langkawi.
Car rental Langkawi Airport
You can find counters of car rental companies near the exit of the Arrivals Hall. There are only two rental companies present at Langkawi Airport, both local providers, Island Point and Kasina.
Taxis are the only means of public transport at the airport. There's a kiosk (Tel. +60(0)4 955 1800) outside the Arrivals Hall where you can book a taxi and pay. There are fixed rates, which by the way are rather high. Some examples: to Pantai Cenang 24 ringgit, to Pantai Kik or Teluk Datai 28 ringgit, and to Tanjung Rhu 35 ringgit.
There's a short and long-term parking lot in front of the terminal.