1

The cheapest way to get to most places from Bangkok is by bus, but make sure you get a government one from a bus station rather than the ones offered by travel agencies around Khao San Road. Avoid those at all costs - they may seem like a good option but stories of theft and other issues are rife.
Marginally more expensive but a much more pleasant way to get down to Phangan is to get the overnight sleeper train. You can buy a combined train - ferry ticket from Hualamphong train station. Go and book it on your first day in Bangkok. 2nd class fan is the way to go - a/c is too cold IMO.

From Phangan it's straightforward to get a ferry - bus ticket over to Phi Phi or Phuket from any travel agency. Then you could fly up to CM from Phuket with Air Asia (book as early as poss) and then get a bus or train back down to Bangkok.

All the train info and timetables are here.

2

Yes, learn very quickly that BEST and CHEAPEST are mutually exclusive. The best is by air, the cheapest is by backpacker KSR buses, where there's a very high risk of your being robbed. The compromise, and most widely used - the train network and the nationwide system of Govt buses.