Newport National, as mentioned, is a good bet. Reality is, with the public golf available on the island Newport National is the best by a mile. A few goofy holes but otherwise a good course. When I lived in Newport (12 years ago) Green Valley in Middletown was a good budget alternative with some really interesting holes. Montaup further up the island also has some good golf. Depending on your access there are obviously some good private course choices up there. I wouldn't try and venture too far off the island during the summer ,traffic can kill a day for you.