ChickenCanoe is right on! Maine has it all. Mountains, canoeing on GORGEOUS inland rivers and tidal estuaries, vast forests, small but dramatic mountains. You can go antiquing, on whale watches, camping, hiking, everything. If you time it well, you can take your kids to a pick-your-own blueberry farm and turn them loose with pint boxes. Then ask your bed-and-breakfast owner if the berries could be made into blueberry pancakes.
Being from Massachusetts, I can't avoid pushing my own home state too. We have what Maine has, including rocky coasts (once you're north of Boston), but the mountains are on the western border with New York -- the Berkshires -- and they are lovely and fantastic for hiking and camping. You could start at the border and enjoy the hills, then take the Pioneer Trail east toward Boston (New England Aquarium, Museum of Science -- fantastic! -- Children's Museum, Fanuil Hall/Quincy Market and the Freedom Trail, plus all of the historic communities like Salem (maritime and "witch trials" history and the excellent Peabody-Essex Museum);, Essex, Gloucester and Rockport (gorgeous coast, beaches,whale watching, kayaking and canoeing). Lobstah (we have it as good as Maine!) and chowdah. The only place to avoid during summer is Cape Cod and the Islands, because they are PACKED with tourists and summer folk. Boston and all are busy too, but much better to absorb crowds than the Cape and Islands.
The only "negative" about Maine or Mass. is that the ocean is COLD. Except for the South Shore and Cape of Mass -- nice ocean temps there, but north of the Cape, you could do surgery on your foot after dipping it into the sea.
The rest of New England is also wonderful. New Hampshire has some coastline, but it is crowded; they are better for rivers and mountains. The White Mountain range is amazing and you could easily spend your entire vacation there, canoeing on the Saco River, climbing mountains, camping in the great parks.
I have a soft spot for Rhode Island (Newport is fun and the south coast has great beaches). Connecticut has sandy beaches with warm water, nice state parks for camping, beautiful rivers -- the northwest corner is wild and beautiful and right on the New York border.
Take your pick.