You take a flight from your respective country to Bangkok. From Bangkok, you can take a flight to Trat, then a bus and ferry to Koh Chang. Or you can take a bus or taxi from Bangkok to Koh Chang directly.
You have to first fly into Bangkok from your respective city. Then there is the Bangkok airways that flies three times a day from Bangkok's Suvarnabhumi Airport to Trat Airport. From Trat, there is a bus and then a ferry service to reach Koh Chang that takes 500-900 Baht. The flight takes 2,500-3,500 Baht and 1 hour to reach.
The cheapest bus service is from Bangkok to Laem Ngop where the Koh Chang ferry peirs are, taking 300-500 Baht and 5 hours only. A more comfortable bus service from Bangkok to the Lonely Beach of Koh Chang has started which also takes 300-500 Baht. If you take a flight till Trat, there are bus services from Trat to Koh Chang taking 500-900 Baht. Bus is a much cheaper option for those who don't want to spend on a flight ticket.
Most ferries operate from Laem Ngop which has 3 piers. Koh Chang has 2 main piers Tha Dan Kao, which is 45 minutes away from Laem Ngop, and Tha Ferry Dan Kao, which is 1 hour away.
Songthaews and taxis are the most popular modes of transport in Koh Chang. These cost 50-150 Baht per person depending on the distance. There are motorcycle and motorbike renting shops for 150-300 Baht per day. Hitching is also very famous in Koh Chang, many people often offer rides. This is a much cheaper option compared to others.