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 y ... Read more »