The fastest way to get to Koh Chang is to take a Bangkok Airways flight to Trat province, then use their transfer service to the pier in Laem Ngop for the boat to Koh Chang, and on to your hotel. At last check, there were two flights a day between Bangkok and Trat. The pier is only about 20 minutes from the airport, and transfers are available from Bangkok Airways directly. The cost is 300 to 370 Baht (7.68 USD to 9.47 USD) per person and vans run every 30 minutes from 6:30 to 18:30. Note that there are no taxis at the airport. You must either use the airport transfer service, or arrange for a transfer through your hotel.
The longer, cheaper route is by bus to Trat from Bangkok's Eastern Bus Terminal on Sukhumvit Road. The trip takes at least six hours, so it's an all-day jaunt to get to Koh Chang by bus. If taking the bus, on reaching Trat you will need to catch a songtaew to the pier at Laem Ngop. There are boats to Koh Chang about every half-hour during the high season from November to March.