I'd suggest not trying to do too much travel in the limited time you've got. I might recommend as follows:
- Spend two nights in Bangkok on Khao San road just enjoying the scene. Go to a good tailor on your first day and order some suits and shirts for pick up when you come back from Bangkok at trip's end.
- Head to Koh Phangan by either nightbus to Surat Thani or plane to Koh Samui and then ferry from either to Koh Phangan.
- Enjoy the full moon party and the island generally for about 5 days before heading by ferry to Koh Tao.
- Spend 4 days on Koh Tao getting your open water diving certification and enjoying the parties.
- Make your way back to Bangkok and spend another night or two there. Pick up your suits and shirts and head home.
There's lots more to see and do in Thailand but for your first trip and the length of time you have, that's about as good as you can do.
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