Walk out the front door, make a right, walk over the highway on Atwells Ave. and you're a couple of hundred feet from Federal Hill, the heart of Italo-American Providence, where there are many to choose from. The closest to the hotel will be Camille's and Mediterraneo on the left and Angelo's on the right, the first two a little more upscale than the latter, but all acceptable. Further along on the same side as Camille's you have Siena, which has a "younger" atmosphere and more of a bar scene, but is popular on this board. Pane e Vino also has its fans, at least in part for the impressive wine list; it's on the other side, and is more or less the end of the restaurant strip. More views can be found in this thread http://chowhound.chow.com/topics/818282 or by searching under "Federal Hill" or any of the resto names I listed.
Wherever you eat, skip dessert and head to Pastiche on Spruce St (parallel to Atwells but one block north) for coffee and sweets.
None of this needs a car or cab unless the weather is really foul.