The only way to reach Koh Samet island is by a ferry. So you have to reach Rayong by road from Bagnkok first, and then take a ferry to Koh Samet. This island is a part of Khao Laem Ya-Mu National park, hence there is an entry fee of 200 Baht for foreigner adults.
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How to reach Koh Samet by flight
You have to reach Bangkok from your respectiv ecity first, then reach Rayong by a cab or bus, and then take a ferry to Koh Samet.
How to reach Koh Samet by bus
You can take a bus from Bangkok's bus station to Rayong or Ban Phe which takes 3-4 hours, and costs around 200-400 Baht. After which, you have to take a ferry to reach Koh Samet.
How to reach Koh Samet by Waterways
After reaching Rayong from Bangkok, you have to take the ferry from Ban Phe pier to Koh Samet costing anything betwee 40-60 Baht. There are ferry boats every hour. Speed boats are expensive costing around 200-400 Baht, but takes only 10-15 minutes to reach.
Local transport in Koh Samet
There is only one main road, which is a little rocky, in Koh Samet. Walking around is the best way to explore. Other options are taking a songthaew costing 600 Baht for a private trip, and 50 Baht for shared, or renting a scooter for 200-300 Baht per day.