You could hire a car and just explore locally. The driving isn't so bad as long as you take it slow, keep your eyes open all around you and expect the unexpected...
Some accurate, to scale maps here to help you get around...
Punta Cana Map | punta cana tv
Macao beach is worth a visit. Huge, beautiful wide bay, surf to the left, calm water to the far right. There's a rustic, inexpensive seafood shack at the far right of the bay - cold beers, fresh fried fish... on Map 2.
You could take a drive into Cap Cana (10 mins south of the airport) see Map 8. Wander around the deserted, luxury marina or take a walk on stunning Juanillo beach. Or drive up to the restaurant Faith Farallon which sits 265ft above sea level on a cliff - nice place to have a drink and take in the views. Everything is signposted in Cap Cana and it's perfectly safe to drive there.
At the security gate just tell them you're going to the marina and they'll let you through (good idea to have some ID on you just incase).
Another beautiful beach to visit is the one at Puntacana Resort & Club - Playa Blanca (Map 7). At the security gate tell them you're going to Playa Blanca and they'll give you a plastic sign for your car (again have some ID on you). The Segway tours run from there so you could give them a go. They'll take you all around the grounds and you'll get to swim at one of the beaches and in the crystal clear freshwater lagoons (fed by underground springs). Also worth a visit is the Ecological Foundation inside the same grounds - where you can buy organic, forest honey.
btw: not much in the way of road signs in Punta Cana. And, like everyone has said, plan every journey so you won't be on the road after 7pm when it gets dark.
Have fun!