Stone Town is for walking. The lanes are too narrow for cars and most are pedestrianised by default — a few bicycles and motorbikes will squeeze past you. The whole old town fits inside about a kilometre square and there is no part of it more than a fifteen-minute walk from anywhere else.
A simple printed map will help for the first day, after which you will accept that getting lost is the point. The four landmarks on the seafront — the Old Fort, the House of Wonders, the Sultan's Palace and Forodhani Gardens — are the easiest reference points; if you head west toward the sea, you eventually hit one of them. Note the carved doors of any place you stay, and look for them on the way back.
For trips out of Stone Town — to the beaches at Nungwi, Kendwa or Paje, or to the spice plantations — agree a half-day or full-day taxi at your hotel. The standard local minibus, the dala-dala, runs out from Darajani for under a dollar to anywhere on the island, but luggage and timing make it a stretch for most visitors.