There is a bus service but most hotels have their own ground transportation. For getting around Dubai there is a fairly good bus service, mini buses and taxis. Most people who are staying for some time make sure they have access to a vehicle, as there isn't a reliable public transport system beyond the city. Car hire is not too expensive, the roads are in good condition but the indigenous driving population makes up the rules as they go along. Be very careful until you are used to it.
Roundabouts are particularly hazardous as are wandering camels.