Although I would recommend a drive to Cape Cod, I wouldn’t recommend it on a weekend in the summer. The bachups for the Sagamore Bridge just getting onto the Cape are horrendus.
Do go to Franconia Notch. Even if you don’t take the challenging hiking trails, you can see the Flume, Houston Rock, Cannon Mountain, the Whirlpool, and The Place Where the Old Man of the Mountain Used to Be. You can also go to the restaurants and shops of placwes like Lincoln, NH.
Gas prices in Connecticut are way higher than in Massachusetts.
I’m partial to the North Shore, but I think you really have to go see Salem, Massachusetts. There’s a lot of preserved (and largely forgotten) history, the new Peabody-Essex museum is incredible, you can see places where Nathaniel Hawthorne worked and wrote about (House of the Seven Gables) and the hoopla surrounding the whole Salem Witchcraft stuff is not to be missed.
Do see the Freedom Trail in Boston – well worth it, and it’ll take you through downtown and past a lot of histoiric sites.
Despite all the changes recently wrought (most of the used bookstores – and some unused ones – are gone! Sob!), Harvard Square and Back Bay Boston are great walks through eccentric neighborhoods and odd shops.
See Mark Twain’s House in Hartford. They just added a museum!
See at least one “restored village” in NE – Mystic Seaport in Connecticut(19th century whaling village), Old Sturbridge Village in Sturbridge, MA.(19th century America), Plymouth Plantation in Plymouth Mass.(17th Century American settlement), Old Salem Village (several ages, and often neglected, in Salem, MA). Or see one of the Shaker Villages (Sabbathday Lake, Maine – the only one still active; Enfield Village in Enfield, New Hampshire; Canterbury Shaker Village in Canterbury, NH; Hancock Shaker Village in Hancock, MA, in the Berkshires (with its round stone barn), or Fruitlands Museum in Massachusetts, with bits of a Shaker Village, Indian stuff, and the remains of the Utopian Community started in the 19th century.