Do you only want to see beaches? You may want to give the north part of the country a try as well, up around Chiang Mai, for example. It's very mountainous and beautiful.
How are you planning on traveling out from Bangkok? Between Bangkok and say, Phuket, there is not much to see and by bus it must be at least a 14 hour trip or so. Your island destinations will really depend on whether you like laid back chillout or crazy partying. I would suggest researching islands, looking at a map, figuring out how each island can be reached, and then plot a route. For example, some islands can only be reached from the mainland via ferry. Some, like Phuket and Koh Samui you can fly to from Bangkok. Some islands can only be reached by ferry from other islands. I am not trying to make this sound confusing, but it is not easy to recommend a 'route' per se because some islands are off the east coast, some are of the west coast, some are farther south than others. Some of the popular islands are as follows:
Phuket (big island, good mix of laid back or all out partying depending on local)
Koh Samui - party central for the Euro crowd
Koh Phangan - famous for full moon parties
Koh Samet - can be reached via a 2 hour bus ride from bangkok, plus a ferry ride.
Koh Phi Phi - reached via ferry from Phuket
Koh Chang - reached via ferry from mainland
Koh Tao - I think reached via ferry from Koh Samui but I could be wrong
There are others, but this is a starting point for you to do some research. Good luck.