Bangkok to Hat Yai to Koh Lipe
The quickest way is to fly to Had Yai, take a taxi or minibus to Pak Bara, from where you take the boat to Koh Lipe. If you arrive in Had Yai on an early flight, you can make it to Koh Lipe in 1 day.
You can also take a train or bus from Bangkok or Kuala Lumpur to Had Yai, but it is a very long trip, and often not much cheaper than a flight.
You will have to go trough immigration on both sides as this is an international border crossing.