Belize City is small enough that the Fort George area — where most of the sights and better guesthouses are — can be covered on foot in a day. The streets in Fort George are generally safe during daylight; the Southside (south of Swing Bridge and away from the waterfront) is less so.
Taxis are the standard way to get around the city and are inexpensive. They don't use meters — agree on a price before you get in. Most trips within the city centre should be a few dollars.
Cycling is technically an option but the city is not well set up for it and the traffic can be hectic. Walking the waterfront is pleasant; walking into residential neighbourhoods alone at night is not recommended.
For day trips to the Belize Zoo or cave tubing, the shared bus system works but is slow. Most travellers take organised tours or arrange private transport through their hotel.