There are regular songtheaws rom 07:30 to 18:00 between Kata Yai and Phuket Town and the fare is 30B. The route starts at the southern end of Kata Yai, runs along the front road, over the little hill to Karon, then all the way along the beach there, before taking a left at the roundabout where it heads inland and doubles back to Phuket Town. If you're heading to either Kata or Karon Beach you get the same songtheaw and they can also be used to switch just from one beach to the other.
Expect to pay an extortionate 300-400B for a tuk tuk to Kata from Patong. If coming from the airport, you can expect to pay 750B for a taxi or around 200-250B for a seat in a minibus.