When you get to Bangkok you maybe tired and exhausted after the long flight, and most people want to spend the first night in a hotel in Bangkok. Finding a place that fits you shouldn't be a problem (travel guides or the internet can be helpful).

There are about 350 km from Bangkok to Koh Chang.
You can go by plane to Trat Airport (they got a shuttle service to your resort on Koh Chang), take a bus from Bangkok to the ferry pier, or use mini-bus or private mini-bus directly from Bangkok to your resort on Koh Chang.
Nearly every small tourist agency in Thailand can book flights and bus-tickets to Koh Chang.

On Koh Chang, the taxies awaits you at the exit of the ferry.
Just tell the driver you want to go to Lonely Beach at the 7-day minimart. The Lonely Beach Resort is located at the opposite site of the street.

It is also possible to make special arrangements with Lonely Beach Resort to get picked up at the airport or at the ferry.