Where are you staying? Kids in tow or just adults?
Do you like wine? Try Sakonnet Vineyards.
Narragansett shoreline
As you mention, Newport Mansions is good. Can't believe they were just summer cottages. Also there, Cliff Walk, "10 mile" ocean drive, Fort Adams.
Foxwoods/Mohegan Sun Casino
Old Mystic Seaport, Mystic Aquarium
Attend a Waterfire In Providence (Check Schedule)
Patriot Place (If you are staying at Normandy Farms)
Places to eat.
Iggys Doughboy Doughboy and Chowder House
Wrights Chicken Farm (If you head North of Providence)
Chelo's (Small RI only chain)
One of the many seafood shacks. Local to me, Flo's and Evelyn's comes to mind.