Phuket is not a small island, like phi phi. It is crowded, and less exotic island too! You still have less crowded parts, like New Harn beach, but then you are further away from "shopping areas"!

Now, from your description, I would say to avoid at least "Patong", that is very crowded and a city by the sea!

Kata, has a nice beach, and several lodging options in every category. Tuk-tuk, is a problem in Phuket, it is very expensive! Around 400 baht from Kata, to Patong for shopping, one way, only 15-20 minutes ride!

A hotel is difficult to suggest. Just have a look at sawasdeedotcom. Very good description of hotels. Not that much choice, on the beach!

Although it has become rather touristic, Phuket still has some things going for it. You still have nice, not that crowded beaches, with a lot of choice of accommodation, and if you want some modern amineties and shopping, it is all there!