As you can see in the photo, locals use all kind of means of transportation. The center of Aswan is small enough to be visited on foot.
You will need to take a taxi to visit some of the monuments in or near Aswan. Taxis are really inexpensive here, as in the rest of Egypt. You do have to negotiate the price, though. Before you stop a taxi, you need to know the price of the ride. Taxis have no taximeters, and the ones that do have it, don't use it. You must know that tourists always pay an extra price. How much this extra is, depends on you bargaining abilities.