I quite like Phuket. Three places we've stayed are:
Baan Krating, right down south, near Nai Harn beach - it was a good place to stay, very peaceful, but maybe too peaceful for teenagers.
Peach Hill in Kata - this was OK - no sea views but we managed to get a bungalow which is what we like. Handy for Kata (north Kata) and tuk tuks right outside to go anywhere you want.
Coral Island resort - as the name suggests, on coral island. You have to get a boat there and there is only this place to stay. Again, it may be too quiet for teenagers.
All of them we found on the sawadee.com website.
But I'd look at staying in Kata for convenience and nice beaches. Patong is too noisy for me and too seedy, but your kids might enjoy it being a bit older - I went to Amsterdam with my family when I was 15 and thought it was brilliant