What are the cheapest and best transport options from Patong to Ao Nang and from Ao Nang to Koh Lanta?

Use this:

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Cheapest way would be buses and song teaws.

Patong to phuket market by song teaw

Song teaw to Phuket bus station number 2

Bus to Krabi, Then Song teaw to Ao Nang.

Ao Nang to Koh Lanta

Song teaw to Krabi Bus station bus to Koh Lanta.

3 other ways.

1) Go see a travel agent in Patong (if there is any left) and get mini bus to Ao Nang from Phuket. Most mini buses depart from bus station number 1.

2) Ferry from Phuket town. No idea of schedules before could not go directly usually ferry to Phi Phi first then another ferry to Ao Nang or Railey

3) Use one of the apps grab etc

No idea of costs now been a while since have done.

Most adventurous and fun way is bus and song teaw no good if you are carrying a lot of luggage though.