We usually recommend tourists to visit one of the several "Legal Seafoods" when in Boston as fish is a big local food and they do a great job with it. If you are not from one of the coasts, you may get a kick out of our North End, where we have authentic Italian people making authentic Italian food - they are all good. (If you are from Chicago, NYC, Philly or San Fran, you will have a lot of great Italian Food at home, so stick with the fish).
I see you are into "Romance" so if you are out with a Date Aujord'*** at the 4 Seasons is a long classy meal. If you are looking for a cool scene and a great meal, try Mistral. If you are looking for a cool scene, try Sonsie.
I suppose if you are going out specifically for Lobster, you should ride over to Cambridge to Jasper White's Summer Shack, I know a Chef who had the dinner the night before his wedding at this place as Jasper is the guru of lobster. It is NOT fancy, lobster is not supposed to be fancy (remember Flashdance!).
Grew up in Boston and have eaten at just about every resturant. The No Name restaurant is a must. It is a "salty" seafood restaurant on the waterfront with a great history. Seafood is awesome and the seafood chowder is the best I ever had. Very casual and famous for its blue collar appeal. Anthony's Pier 4 is fancy and wonderful. Fresh popovers delivered to your table with top notch food and incredible views. Go to the North End and just about every Italian restaurant is great. My favorite is Trattoria Scalinatella. Legal Seafood is great and the service is top notch. The melting pot is a great fondue restaurant. Top of the Hub for a pretentious but wonderful dinner at the top of the Prudetial Center. Fanuel Hall for great pickies and desserts. Here is the best website for reviews, prices and directions. http://calendar.boston.com/restaurants
I was born in Boston. When I have visitor's from out of town come, here is the culinary trip I take them on....
Start early with a pint and a bowl of chowda at The Union Oyster house. It is the oldest restaurant in America. Nothing fancy, but the history makes for a good story, and the chowda and cornbread are fantastic!
From there we wander around the Fanueil hall area for some sightseeing...and to build up the appetite. Perhaps even sneak in another pint or two at any one of a 100 pubs or taverns.
For Dinner it depends on the taste of my guest but here are can't miss choices by category....
Italian: many to choose form in the North End, but my favorite Italian meal is not located there, it's Sorellina's near Copley Square.
Steak: Grill 23 & Bar in the Back Bay. It doesn't get any better than this place for beef!
Seafood: Can't go wrong with Legal Seafood. It's a chain but it still does things right.
If you can still stand finish the evening with a cocktail at the Top o the Hub!