I went in August - yes, its the rainy season but that side gets protected from the worst of it. There will be nice days and some wet days. Seemed to rain mainly at night, followed by a nice sunny day.
Koh Samui is busy nowadays - 10/15 years ago it was v.quiet. Koh Tao is what Koh Samui was 5 years ago and Phangan is what Samui was like 10 years ago, so depends what you are after.
As for flights, I cant recomend any airline, but lots of budget airlines have opened up in the last few years and do try to book it before you go, otherwise, you have to go to a Thai agent and they will mess you about with putting you on waiting lists etc.
There is a train from Bangkok to Surat Thani - followed by a boat ride to the islands - only for the brave, as its a long journey, but certainly an experience that would stay with you.