I've been there about 8x, starting when I was 10. Its very family friendly and great for kids.
Phuket has an international airport, so there is the option to fly directly there and fly out from Bangkok. I agree with another poster that Phuket is touristy, but there are less touristy beaches to stay at and lots to see/do. I would avoid staying in Patong beach though! Full of girlie bars and (mainly) western men walking around with much younger Thai girls. The last time we went to Phuket, we stayed down the coast at Kata beach. Much quieter than Patong and more family friendly.
Koh Chang was much quieter and very relaxing. Our hotel has spa baths on the balconies, so you could soak and watch the sea. I recall we had to get a ferry to the island, whereas Phuket island has a bridge.
From both Phuket and Krabi, you can get a ferry to go to Koh Phi Phi. I think they have closed or now limit the number of people that visit Koh Phi Phi le- where they filmed 'The beach'. DH and I scuba dive, but there was great snorkeling too.
What time of the year are you going? Between Jan-April, avoid Chiang Mai. They burn off crops and the air quality is extremely poor and hazy.
Internal flights used to be fairly cheap and easy to book yourself. Nok air, bangkok airlines and Air asia being some. Also considering flying into another country nearby, then taking a short flight to Thailand. Air Asia used to have some great deals from London to Kuala Lumpur, then a cheaper flight to Bangkok, Phuket. Krabi or Chiang Mai. You'd need to check the luggage allowance though because they often differ.