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.
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.
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.
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.