Phuket Old Town can easily be visited by wlaking. The most interesting streets are Thaland Road, Krabi Road, Dibuk Road, Phang Nga Road, Yaowarad Road, Ranong Road and Ratsada Road, and ofcourse Soi Romanee (in Thalang Road). In general, people start the visit by the Chartered Bank at the intersection of Road Phuket and Phang Nga Road and walk in Phang Nga Road to discover the On On Hotel (where Leonardo DiCaprio stays at the beginning movie "The beach "), before taking Yaowarad Raod (turn right) to Thalang Road and there you are free to choose where to go but I suggest you do a loop and go to Dibuk Road (3rd from left), then take Satun Road (turn left), then Krabi Road (turn left) where you can visit Thai Hua Museum, then go straight into Thaland Road remembering to go look Soi Romanee.
From the bus terminal number 1 in the heart of the city, it takes only a small 15 minutes walk to reach Phuket Old Town. Note, if the experience tempts you, that there are some guesthouses in old houses in the neighborhood. The coolest is probably Old Town Hostel that offers double rooms and dormitories. Less traditional, a 1-minute walk from the old town, the Quip Hostel with a roof bar and comfortable beds is a good option (you can hear the traffic, earplugs can be useful if you're a light sleeper).