Phuket Town is located on the east side of the island, it's just been upgraded from Town status to City, as it's been growing quite a bit over the years. Here you can visit the usual shopping centers, walk around enjoying the still town like life, traffic can be a bit hectic during rush hours (school start / school end). But if your on a motorbike traffic jams is nothing to worry about. You should visit the famous old part Phuket town, and have lunch at one of the many smaller restaurants in the area.
Phuket Town attractions are mostly about having a look at the history and life of Phuket including markets, temples, old houses, shrines, museums, and what probably is more fun for the kids are the Phuket Butterfly Garden & Insect World, Phuket Orchid Garden & Thai Village, The Monkey Hill. Some interesting spots are The Amulet Market in Phuket Town, Baan Chinpracha, Old Phuket Town, Khao Rang or Rang Hill.
There is plenty of Chinese Shrines to visit for those interested, just to mention a few : Bang Niew Shrine (Old Phuket Town), Hok Nguan Kung Shrine (Surin Clock Circle), Jao Mae Kuan Im Shrine (Bangkok Road), Ju Tui Shrine (Ranong Road), Kiew Tien Keng Shrine (Saphanhin Park), Lim Hu Tai Su Shrine (Samkong), Sapam Shrine (Sapam Village), Yok Ke Keng Shrine (Samkong), Shrine of the Serene Light (Old Phuket Town).
If you are interested in museums or old hoses you should visit Phuket Philatelic Museum, Thai Hua Museum, Thavorn Hotel Lobby Museum and Baan Chinpracha. If Buddhist Temples are what you want to see then you should visit the following two Wat Kajonrangsan and Wat Putta Mongkon.
If you are in the need of some shopping, apart from the usually super shopping centers such as Tesco Lotus, Big-C, Central Festival, Robinsons Department Store, Ocean Shopping Mall, you have the smaller and more interesting shops to walk around in. The big Pensri Market and Expo Phuket is also amazing places to walk around and browse all kind of stuff from jeans, trendy clothes, toys, games and the list never ends, and at amazing good prices even if the quality tends to follow the price.