Sparks on Broadway. Creative , eclectic, excellent quality. By no means run of the mill-more hip than fancy. Tuna tartare pizza, crab succotash,fried oysters with chipotle aoli- you can't go wrong
I agree with Mama Louisa's - the food is fantastic! Go to the Castle Inn for drinks just before sunset an then drive back into town and have dinner at Mama Louisa's - great night! Have fun!
I have to say everyone who responded here has hit the nail on the head they are all great and the nice thing is that when you have finished dinner you can stroll through the streets and go window shopping.
If you're looking for a quintessential Newport experience (i.e ocean view, luxurious surroundings) then Castle Hill would be my pick in terms of food and locale.
If the view doesn't matter Tucker's is great (or at least used to be - haven't been in a year or so).