You may want to consider going to Koh Lipe or Koh Phi Phi.

I went to Koh Lipe on my last trip and loved it. I stayed at Sawan resort which is probably my new favorite beach resort. If you are looking for something less crowded and less tourist then I suggest Koh Lipe.

Here's a picture from the balcony of the bungalow I stayed in on Koh Lipe in September:

If you want something more touristy but much less touristy than Phuket then consider Phi Phi. Koh Phi Phi has way more accommodations and more to do. You can purchase island tours, snorkeling excursions for cheap. Scuba is also available. For the nicest and clearest waters, you'll want to go between November and April. Rainy season is less crowded and resorts are cheaper for obvious reasons.

For Koh Lipe, I flew into Hat Yai airport, took a van to Pak Bara dock (2 hours), and a boat to the Island (hour and a half).

For Koh Phi Phi, you'll need to fly into either Phuket or Krabi airport, take a van from the airport to the docks, then a Ferry to the island.

For cheap flights throughout Thailand, consider Nok Air, Air Asia, or Thai Lion Air. You can typically find one way tickets to a lot of places for $40-60 bucks, especially if you book at least a few weeks in advance.

I recommend Agoda.com for all reservations in Thailand. They have very detailed picture, photos, and reviews, and I've never been disappointed by a place that received high ratings on Agoda (8+ ratings is usually what I look for).