These aren't necessarily historic but I'd say they're all classic: The Burren at Davis Square, Bartley's in Harvard Square, East Coat Grill at Inman Square, The Field in Central Square, Legal Seafoods at Kendall Square, Jacob Wirt's by the Boylston stop, the Oak Bar at Copley Square, Bukowski Tavern by the Hynes Convention Center stop, and the Publick House at Washington Square in Brookline. A fair amount of beer spots, but this seemed ok for three guys visiting.
How about getting a drink at the Last Hurrah in the Omni Parker House (the food is average; the drinks and ambiance are great), then going to Silvertone in Downtown Crossing for dinner, then continuing on to Jacob Wirth in the theater district for more drinks in a classic old Boston watering hole?
If you go on Friday, you'll get to do the weekly singalongs with the piano guy at Jacob Wirth. It may sound unappealing, but by the end of the night it's usually pretty wild and crazy (especially if the Celebrator Dobblebock is flowing).
I agree with MC; if you REALLY want to do an "only-in-Boston" dining experience, chow down the way a lot of us do here: at our local pubs and little neighborhood eateries. Anyone could eat at Durgin Park (and barely live to regret it). Become a local for a few days and have a Toastie at The Squealing Pig in Jamaica Plain, or a Neopolitan pizza at Antico Forno in the North End (or at Regina's, which can certainly be a tourist trap, but the pie IS damn good. Just don't eat at the Quincy Market location. Only the North End original is worth the effort). Have a burger at R.F. O'Sullivan's, mac and cheese at Silvertone, MORE pizza at Santarpio's in Eastie, a lobster roll at Belle Isle. Join the line at lunchtime in the North End for a slab of square Sicilian pizza and a paper cup of wine at Galleria Umberto, grab a freshly filled cannoli at Modern Pastry, a rack of pork spare ribs at Uncle Pete's Hickory Ribs.
You can look in any guide book and find a decent variety of fine dining places, or talk to any concierge to "discover" the best places. But if I were you, I'd explore this site for all it's worth, and TRULY discover how we really Chow here.
I only wish Tim's Tavern were still around. To me, THAT was Boston.
For historic fairly pricey drinking, the bar at the Parker House/Last Hurrah downtown.
Somewhat historic and drinks with a view, Top of the Hub in the Pru.
For historic well priced drinking and fair eating, Doyle's in Jamaica Plain is definitely fun. Famous politicians went there, interesting space, lots of beers, people say the pizza's good. Burgers aren't.
Wally's on Mass Ave. for drinks with jazz.
Otherwise, checkout the above recs for good drinks and food in unhistoric places.
Locke-Ober is definitely a fine suggestion for an only-in-Boston experience. That Brahmin vibe is unique to the city, and the food, while pricey, is much improved since it was bought and renovated a few years ago. Try the roast beef hash with poached eggs at lunch (very Yankee, in the New England sense of the word), or the lobster stew any time. A bit stuffy.
The North End can be touristy, but with some guidance on this board as to where to dine, it can also be a lovely, charming evening out with great food. We're not the only North American city with a strong Italian-American community and restaurant-heavy neighborhood, but ours can be one of the more Chowish ones, if you know where to look.
I don't think many North American cities have our quality of Portuguese and Brazilian retaurants in one neighborhood, as we do in East Cambridge. These cuisines are wonderful, pretty reasonable, and show off our local seafood bounty well.
Don't waste your time in Faneuil Hall/Quincy Market looking for good food. This is a hateful tourist trap, dining-wise; ditto for the Cheers bar in Beacon Hill.
While they aren't especially unique to Boston, I tend to bring/send visitors to my own personal favorite bars-with-good-food, where they can see a nice cross-section of actual locals while getting a drink or three and a nice, not-too-expensive meal. These include the Franklin Cafe and Anchovies in the South End; Audubon Circle and Washington Square Tavern in Brookline; the B-Side Lounge, Middlesex Lounge, Miracle of Science, River Gods, Chez Henri (bar only), and Green Street in Cambridge; dbar, Blarney Stone, and the Ashmont Grill in Dorchester; Silvertone near Downtown Crossing; Santarpio's in Eastie; the New Bridge Cafe in Chelsea; Mission Bar & Grill in Mission Hill; Zon's and Doyle's in Jamaica Plain; and R.F. O'Sullivan's in Somerville (for burgers and beer).
If you want historic, classic places there are three:
The Union Oyster House near Fanuel Hall. The oyster bar is terrific. Have a beer and some raw bar. Forget dinner there- not so hot.
Durgin Park- great chowder, boston baked beans, prime rib, indian pudding.
Pizzaria Regina- great pizza.
Another one but expensive is Locke Ober- definitely not cheap.
Hope you have a great time.