If staying in Kamari for a week a rental car, scooter or quad would be advised for about 3 days. Most companies offer discounts for rentals of this period. While scooters and quads are very popular, the vehicle best suited to exploring the island in full is a small 4x4 like the ever-popular convertible Suzuki Jimny. For the even more adventurous there are street-legal beach buggies that seat 2 and look like a lot of fun.
There is ample parking by the beach, with several larger car parks so don't bother with street parking