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.
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.
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.