Best and cheaper way to get taxi in Phuket and Thailand is Grab. Grab is an application, like Uber ( may be you know) , which yo can help you to pick a taxi with cheaper price.  You can estimate cost of trip before order. 
You're simply should type where you going to move. Also you should  choose wich pay method better for you cash or card. I like a card, this is second reason because I use this applications ( first reason is a price ofcourse). You can use free wifi  in airoports (Suvarnabhumi and Phuket - wifi is available) to use Grab.
You can download Grab here 
 If you want meet your personal driver in airport with your name in his hand,  you should pay your attention KiwiTaxi.
KiwiTaxi is a online taxi service wich provide transfer from many airports around the world. It's more cheaper than in you pick you taxi by yourself. You can fully prepaid your trip  on KiwiTaxi web site. 
I use it when I travel with my family 
( of couse if I travel alone I don't use taxi ) 
Uber 
May be you still don't know about Uber. I use Uber in Bangkok. And also at home - in Moscow. I think it’s better applications for taxi. But Uber is not availlible on Phuket, and also I always have a problems with Uber in Almaty ( Kazakhstan)