If you like calm, you'll love Little Compton. It's a scenic 40 minute or so ride from Newport. Driving up Aquidneck Island on 138 or 114 is the only downside of this side trip, but it's soon over, you cross the Sakonnet Bridge and head down 77/177 (south). The further you go, the more beautiful it gets. Lots of farm stands along the way. You can stop at Gray's Ice Cream at Tiverton 4 Corners for coffee ice cream or a coffee cabinet (other flavors available, although most RIers have no idea why), Carolyn's Sakonnet Vineyard for a wine tasting (definitely worth the stop). Take a left on Swamp Rd and aright on South Shore Rd to South Shore Beach. It's pricey, I think $20 week days, but worth it. You park right above the sand. A few steps and you are there. Then walk the beach to the west, to Goosewing, a farm that sweeps down to a beach on the sea. It's the last one left in RI, maybe in New England, and it's beautiful. I think this is the most gorgeous walking beach in the state.
Naturally you may want to adjust the itinerary. Not many people want coffee ice cream or wine for breakfast! And definitely not together.
On the way back follow the signs to the Commons. It's the most perfect little New England village. Commons Lunch is a nice place to stop for lunch. Heading back to 77/177 the pizza place is a good place for "grinders" (hoagies maybe to you).
Another great quiet place is Jamestown, right across the East Passage from Newport. You can take the Jamestown/Newport Ferry there, for $18 round trip, with stops at Rose Island Light, Fort Adams and a couple along the Newport Harbor-front as well. I love breakfast at Slice of Heaven on Narragansett Ave. The Gran Marnier French Toast is amazing, as are the breakfast burritos. They have nice sandwiches for lunch. There are many other places to eat there. I think Trattatoria Sympatico is one of the most romantic restaurants in RI. If you opt to drive to Jamestown you can visit Ft Wetherill and Beavertail State Park. Both are worth a visit. Beavertail has new lighthouse museum that looks very compelling, although I haven't visited it yet.
Bristol is a very nice town, also. I've usually gone there for the July 4th celebration. I'm not as familiar so I will let someone else point out it's must-sees.