A ideas for where to go.
Mystic Seaport and the Aquarium
Newport Cliff Walk
Jamestown Is
Narragansett/Galilee area
Sturbridge Village
On Cape Cod, the bike trail, ferry to Martha's Vineyard, whale watches, Chatham Pier/Fish Market, Provincetown.
Plymouth
Gloucester/Rockport
Newburyport
Kittery
Kennebunkport
Cape Elizabeth
Boothbay
Rockland/Rockport ME area (a very slow drive on US 1, might be okay in June)
Acadia of course, nice bike/walking paths
There are more nice area north of Acadia but it gets fewer and farther between and might not be worth your time. Campabello Is just over the line in Canada is nice, but it's a long drive.
Kancamagus Highway (rt 112), White Mts (Mt Washington etc.)
Ben & Jerry's
Cabot Cheese
Champlain Islands
There's also a ton of stuff in NY state if you consider that part of NE., enough there for another month by itself.