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.
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.