Taxi prices are outrageous in Phuket and while grab(Uber alternative) is available there the prices are 3-4X the prices in Bangkok
Instead consider renting a car, you can find good deal, it’s allows you to explore all the different beaches on the island and best of all many of the large companies include insurance with a zero deductible in the price so you don’t have to worry.
We paid 900thb a day from Budget, so that’s about $29, it included full insurance with a 0 deductible, which is a good thing to have since driving there can be a bit stressful, actually parking is the worst since you just have to basically park on the side of the road. Not a lot of rules, we actually scratched up the car parking one night so we were thankful for the insurance.
In terms of what companies I would generally recommend sticking with the major companies, the prices are not much more then the local places and they just seem better and more trustworthy if there is an issue.
Also most companies will offer to pick up and drop off, we rented from budget and they dropped off at our hotel on the south part of Phuket, like 45 minutes/ an hour from the airport for free the same day we ordered.
One last thing, just because it says there is no availability online that’s not always the case so it’s worth calling the offices directly.