No one from Rhode Island really knows anything about Westerly - it's part of Connecticut, as far as we're concerned. But for Providence and elsewhere...
I second the suggestion of the Modern Diner on East Ave. in Pawtucket, about a mile over the Providence line. While they always have good food, their weekend specials are phenomenal. I'm especially fond of the cranberry almond pancakes, but the polenta with sausage is also great. I believe the Modern Diner was the first diner to be added to the National Register of Historic Places. Definitely a place to hit.
Apsara, just off Elmwood Ave., is great for Vietnamese and Cambodian. I like Thai Orchid, on Park Ave. in Cranston (just off Route 10 a minute south of Providence) for, well, Thai. It's a little hole-in-the-wall with only a couple of tables, the people are very friendly and the food is good. I'm especially fond of their soups. Nick's On Broadway in Providence has a lot of people yammering about it - they do breakfast and lunch, upscale food at downmarket prices. To tell the truth, I thought they were good but not great - I had a pork loin there that was a little tough, and walnut/blue cheese sauce I had in a duck wrap totally overpowered the duck. I'd stick with the Modern Diner.
Caserta's (Spruce St., near De Pasquale Sq. on Federal Hill) is the classic Providence pizza place. To tell the truth, I don't think their pizza is as good as it once was - about ten years ago, their crust changed and became a little lighter and crunchier, not to its benefit, in my opinion. Still, their sauce is great, really zesty, and the Pepper Pig (a pig in a blanket with italian sausage and green peppers) is fantastic. Cheap, too. They have a very limited selection of pizza toppings. Be forewarned: if you order a plain pizza, it will be
plain - just crust and sauce, no cheese. Specify cheese if you want it (although a pizza with just anchovies and sauce is great.) Also a medium is basically what most would consider a large - a large pizza is actually two medium pizzas.
The nearby Angelo's Civita Farnese (Atwells Ave.) is another classic Italian eatery in Providence. Very basic Italian-American food, wonderfully prepared. Get the meatballs and french fries, the ravioli or the roasted chicken. Everything on the menu (listed on a post in the middle of the dining room) is a la carte, so be sure to order explicitly anything you might want.
There are a lot of Italian restaurants and shops on Atwells Ave. and it's a fun place to visit. Further up the avenue, away from the Italian district, is Mexico, one of the best Mexican restaurants in the area. I've never had a bad dish here - the enchiladas are especially good. For more Americanized Mexican fare (and margheritas), Tortilla Flats on Hope St. is the local leader. Since it's summer, there probably won't be too many Brown students jamming up the place. For a great bar, Lili Marlene's on Atwell's is a good choice. It's virtually unmarked - just the name is stenciled on the door. Look for it near the Surgical Supply store (now going out of business) on Atwells - if you pass the church and start heading downhill, you've gone too far. A more boisterous local gin mill with suprisingly good food (though avoid the pasta) is the East Ave. Cafe, right next to the Modern Diner. They do pig roasts every week, I think on Saturday.
The best fish and chips to be had in RI is available about 20 minutes north of Providence in downtown Woonsocket at a place called Ye Olde English Fish and Chips (S. Main St., Market Sq.) It's a hokey name, but the fish is superb. It's not a clam shack but an English-style chip shop (the founder was a Yorkshireman.) Try the fishcakes: a piece of fish wrapped between slices of potato, the whole thing battered and fried together - like fish and chips in one. Really fantastic. They're closed Sundays and Mondays and have limited evening hours, so call before making the trek, if you are thinking about it.
I'm not wild about Gregg's food, but no one could disagree with their desserts. The cakes are terrific, available for takeout or to eat in, by the slice or whole. For other sweets, I like Brickley's Ice Cream on Route 1A in Narragansett. A bit out of your way, but really amazing. The lemon ice cream has the most intense, pure lemon flavor possible and all of their ice cream is extremely smooth and creamy.
Ooh, I should also mention LJ's BBQ, on Douglas Ave. in Providence. Yeah, this isn't barbeque country, but it's a nice joint, the people are great and the pulled pork is very good. They also carry great local sodas from Joe Wold - try the Cream Soda.
Whew! You get the idea. Providence - all of RI, really - is a great place for food, so you shouldn't have any problems. Hope this helps, though.
JL