Aside from that one rather unpleasant incident, taxis are pretty inexpensive and the most convenient way to get around. Generally, there are taxi stands in front of the mall. Taxis in Dubai has a minimum charge of 10 AED regardless of distance. It's even easier to just take the very modern metro in Dubai.
Buses are inexpensive. You'll often see bus timetables taped to the ticket booth. Long distance buses are pretty comfortable, especially if you're a woman. Women get to board the bus first and usually get the section at the front of the bus. Since the mostly male migrant labor force means that there are more than twice as many men as women, men usually need to stand at the end of a long line whereas ladies almost always get a seat in the "ladies only" section.