I rented a car in Phuket last month. While I've not driven in Langkawi, I have driven in other parts of Malaysia and Phuket is very similar - lots of motorbikes.

Driving in Phuket (I'm used to driving on the other side of the road) is easy if you exercise sound judgement and adhere to the following:
a) don't be in a hurry. Stick to the slow lane and don't be pressured to drive fast
b) watch your lane changes, both mirrors as a matter of course, and watch for bikes
c) never make sudden, unexpected moves. Always signal lane changes in advance so others can adapt
d) Use a GPS
e) be aware some right lanes suddenly become right-turn only

I've driven all over Thailand. I find Thai drivers to be generally OK and patient compared to say Malaysia (KL and Penang especially) and Singapore. Thai drivers will let you filter lanes whereas in Malaysia, good luck with that.

You should get full insurance (SCDW) and I would never get a bike in Phuket (hot, little protection from other vehicles). Parking can be challenging in Old Town but you just go around or park a little further away and something will open up. In Patong, park at the malls (Jungceylon, BananaWalk etc).

In short, you need a car in Phuket.