Bus + direct ferry join tickets from Khao San Road
Khao San Road is the backpacker district of Bangkok. You can find many many things including bus + boat joint tickets to the most popular islands in Thailand like Koh Samui, Koh Phangan, Koh Tao or Koh Samet and Koh Chang. Of course, you will be among tourists but if you have decided to stay in Khao San, those buses are very practical !
The more active company in Khao San is Lomprayah but you will find others like Songserm, Thai Sriram or Travel Mart.