I'm based in Phuket and have extensive reports on local diving in this forum. A search under my username should bring them up.
January is the perfect time to dive in Phuket, with the water conditions ideal and the weather hot and sunny
The diving here is both boat day trips and liveaboards; as you want to do day trips, there are plenty of options. There are about ten or so day trip boats operating from Chalong Pier, in Chalong, every day. Day trips are all day departing around 8:00-8:30 and returning around 4:00-5:00, with three dives and breakfast,, lunch, and snacks on-board. The sites are Koh Phi-Phi (one day multiple sites), the Racha Islands (all day multiple sites), and the King Cruiser wreck, Shark Point, Anemone Reef, Ko Doc Mai (dived as a day trip with 3 sites chosen). For big stuff, Phi-Phi and the King Cruiser are very good. There's also a good chance for Leopard Sharks at Shark Point. If there is a boat going to South Tip at Racha Island during your stay, definitely book it as it's the best site for big fish in Phuket (it's only dived a few days a month due to currents and tides).
If you have 3-4 days of diving, you could actually do all three areas and then repeat your favorite! Generally, you book these trips through a local dive shop on the island. The price includes pick-up and drop-off from/to your hotel. For convenience, I would recommend staying in Chalong or Rawai to be close to the pier. However, it you also want nightlife and beaches than Patong or Kata Beach is better. These dive boats will be crowded, with up to 50 divers onboard, so it's definitely a lively atmosphere. The Sirolo Dive boat is definitely the most comfortable, with all the others a step down in amenities but perfectly acceptable.
I am biased, of course, but you can certainly book your dives at Rumblefish Adventure in Kata Beach, the shop I dive with. We also do daily shore dives at Kata if you fancy a break from the boat for a day. Happy diving.