I've rented a car downtown and driven around the island...only takes 45 minutes to travel full length of the one main road...then you turn around and go back. There are stops along the main road, but it's hard to see them and easy to pass them by.
I suggest touring downtown for a bit and then taking a taxi to Paradise Beach. When I was there several years ago...it was a calm beautiful beach, free to visit, w/nice bathrooms/showers, a restaurant and beachside service from the restaurant & bar. You can rent umbrellas/lounge chairs and for $5.00 (then anyway) you could rent any of the water toys and floating mats to lay out in the water. They also had a really large inflatable tower you could swim out to, climb on and jump off of. The kids had a great time and I did too.