In Cancun, you really don't need a car.
Of course, it's more convenient to have your own car so you can set your own times and schedules, but to enjoy this city, a car is not mandatory. Distances are long, and if you are in the middle of a sunny day, walking doesn't seem like a good idea, but the public bus system does work efficiently and cheaply.
There are many towns and cities in Mexico where you won't see any tourists riding in the public buses; in Cancun, it's quite the contrary, as this is an efficient service and very safe. Basically, if you are staying in the tourist strip or island, there are only two ways to move: toward downtown or the opposite direction, which takes you to the aquatic park at the south end of the island.
You don't need to know the routes, nor speak Spanish, to use the buses going south, but if you are planning on going downtown, it would be wise to ask the driver or the other fellows waiting for the busabout which one to take, as there are different destinations once you are out of the touristic zone.
From downtown, you can go north to catch the ferries to Isla Mujeres in Punta Sam, or you can stay in downtown for shopping and dining. Also in downtown, you can get to the bus station and get a trip to Tulum, Playa del Carmen, Puerto Aventuras, or even the airport, all of them cheap alternatives to taking taxis if you are traveling light. Forget it if you are bringing two big suitcases each. This is for cheap traveling around and enjoying the culture.
Both city buses and out of town ones are clean, safe, and cheap.