NYCer first time in Boston
I like Grotto, too, and it won't give you sticker shock. In general, you'll pay more for less at moderate-range Boston restaurants than in NYC. I'd add Benatti and Marco as possibilities. Mamma Maria is large, but consists of five different rooms, and you could ask for one of the smaller ones. It's the best food in quaint surroundings that I can think of. Sorry these are all Italian. I guess Italian and quaint go together here.
Walk the Freedom Trail, ride the Swan Boats in the Public Garden, go out to Lexington and Concord if you have a car, to see the route of the Minutemen, Walden Pond, the Alcott house. (BTW, "American Bloomsbury" is an interesting book about the lives of the famous 19th century thinkers/writers in the Concord orbit). Walk around Harvard Square, see the glass flowers at the museum there(name escapes me). Otherwise, museums in NYC have more and better exhibits than their Boston counterparts. Foodwise, Boston has nothing on NYC. Again, if you have a car and can go north to Essex/Ipswich, you can find fried clams (Farnham's, Clam Box, or Essex Seafood) that ARE better than what you'll find at home.
Actually, I think Boston has some definite areas of advantage over NYC, food-wise.
Most notably, Boston's Chinese food is heads and tails above anything in Manhattan, and only a few places in Flushing approach Boston's best Chinese restaurants. From Fuloon, to Wang's, to Jo Jo Taipei, to Sichuan Gourmet, among others you can't go wrong.
Another important feature is that extremely high level food can be had for much less than the going rate in NYC. I recall vividly last winter eating at L'Atelier de Joël Robuchon in New York on Thursday night and at L'Espalier in Boston on Sunday night. Both were terrific meals. Joël Robuchon was more than twice as expensive, and L'Espalier was unambiguously better. They are different types of restaurants, to be sure, but the contrast was striking that weekend. So consider splurging at some of Boston's best for less than you would at equivalent places in New York.
There are also lots of unique places in Boston, some of which I'll try to mention below.
But to answer the OP's specific questions:
Baraka Cafe in Central Square is quaint and unique, Ten Tables is definitely cozy (and terrific), and Lala Rokh is very cozy (I'm not the biggest fan of their food, although others love it, and I certainly love the atmosphere). Tangerino is cozy also. T.W. Food feels very "Cambridge" (in a good way). Emma's is less expensive, but both quaint and cozy for terrific gourmet pizza. R.F. O'Sullivans and Bartley's are both quaint for burgers. Locke-Ober is stuffy old Boston-Brahmin club, but I love it to death. It's neither low-key nor quaint, but has a very Boston feel that's hard to describe. The Highland Kitchen in Somerville is cozy and fun, with very interesting food. The Franklin Cafe has a similar feel. A less expensive version might be the Delux Cafe.
For brunch, I've had nice times at Gaslight, Aquitaine and the South End Buttery, although for the food alone I'd head to Sound Bites or the Ball Square Cafe in Somerville. For other kinds of brunch, the Bombay Club does a unique South Indian brunch in Harvard Square, and the Taiwanese places (Jo Jo Taipei, Shangri-la, Wisteria, Chung Shin Yuan, among others) do a nice Taiwanese brunch. If you can get a Harvard alum who's a member to take you, the Harvard Faculty Club has a terrific brunch.
A smattering of general personal favorites that are pretty unique: Clio, Oleana, Flour Bakery, Fuloon, Wang's, Emma's, Evoo, Toro.
As for other things to do, I also recommend the glass flowers at the Harvard Museum of Natural History. The MIT Museum. Seeing Harvard and MIT. Walking around the Public Gardens. Walk the freedom trail if you get a chance. Visit the North End. Visit the South End. Visit Dorchester, to try Vietnamese food, Ali's Roti, or just to walk around. Note how Boston has streets that are mobbed with people along some blocks that veer off into side streets that are literally empty in the middle of the day --- the biggest contrast I see with New York. Have fun! =)
There are a couple more areas that Boston is drastically better than NYC as well, that is Portuguese and Nepalese cuisine. For Portuguese try Portugalia or Atasca and for Nepalese Kathmandu Spice in Arlington. I would also find time to fit in Neptune Oyster in the North End, probably our best Seafood restaurant.
63 Salem St Ste 1, Boston, MA 02113
50 Hampshire St, Cambridge, MA 02139
166 Massachusetts Ave, Arlington, MA 02474
723 Cambridge St, Cambridge, MA 02141
Another cuisine that is perhaps better represented in the Boston area than in NY is Portuguese.
The glass flowers, also is expanding to encompass the long-hidden glass sea creatures:
The Isabella Stewart Gardner Museum is worth checking out, it's unique. And, if the OPs love Asian art, the MFA is non-pareil in this Hemisphere that regard, however much we Bostonians take it for granted.
And the USS Constitution in Charlestown is probably the best museum here - unless you are very tall, most people find it fascinating, though we Bostonians take it for granted.
It would help if you can give some parameters: where you will be staying, will you have a car, will you be relying on public transportation, what kind of food or cuisine and price range, and are you willing to drive outside of Boston (if you have a car)? One excellent, cozy (small) quaint restaurant is in the Fenway section of Boston and is considered to be perhaps the most authentic Italian trattoria in Boston, it is Trattoria Toscana and it is also very reasonably priced.