im in thailand right now and i dont know of many direct flight to Phuket. As mushroom said you can get a connecting flight from bangkok to phuket airport, or for the cheaper option. fly air asia to bangkok, stay there for 1 night along Kho San Road (big backpackers place) here you will find so many travel agensies, all of them do buses to Phuket. pick your bus, get picked up the next day, a bus would be about 8 hours maybe?
ill be in Phuket in a couple of weeks actually, maybe next week im not sure, but when im there if i remember ill have a look around at bus/coach prices?
p.s or another way is fly into Kuala Lumpar and get a coach up, but you would have to cross the thai boarder in which case you would only get 15day free visa. if you fly in you get 30days.
anyway, happy planninga holiday!!
p.s ive heard fellow backpackers say Phuket is very touristy, you could grab a ferry to some nearby islands, Koh Lanta is suppossed to be quite beautiful.