Below are some links to 2 area scooter rental shops, Eve's Cycles aka St Georges Cycles, and Oleander Cycles. Eve's Cycles also has an airport location so you can fly and ride right away!
Both companies offer bicycles for those who want to see the island on a slower places. Both will give you instructions; but you must know how to ride a bicycle plus have a valid driver's license in order to rent.
We strongly recommend that you DO NOT ride during rainy weather, as Bermuda's roads can be quite slick; and you can get a case of the dreaded Bermuda Road Rash.