Bubbles suggestion is good so I will throw an alternative idea out there...
Arrive NYC, stay there a week. Make sure wherever you're staying has air con as the city in August can be miserable.
on the 7th day, fly to Burlington VT, rent a car where you pick up from Burlington and drop off at Logan airport in Boston in 3 weeks time. Stay the night in town, have dinner at Hen of the Wood on Cherry St (book in advance!!)
Now do some driving. The back roads in New England are lovely, loads of places to stop off for ice cream, farm stands, random roadside attractions, trading posts, etc. Avoid the highways and get your teens to spot stuff that looks cool out the windows, pull over often in little towns, for coffee, ice cream, instagram moments...
Day 1: North to Smuggler's Notch, Jeffersonville, and pick up Rt. 100 and follow it back south through Stowe, Waitsfield, Warren, and then pick up Rt. 4 through Woodstock to Hanover. Stay the night in Hanover.
Day 2: visit Dartmouth College in the morning, then either drive up through Franconia Notch to the Mt. Washington Hotel, or drive over on Rt. 25 through Wentworth and Rumney to Wolfeboro on Lake Winnepesaukee.
Day 3: Spend the day in/on/around the lake. Go to the Wright Museum of WWII (the one with the tank crashing through the wall). Swim in the lake which is lovely fresh water. Dinner in town. OR if you went north to Mt washington, go hiking, and stop at Clark's trading post in Lincoln.
Day 4: Now you have a choice, you can drive a long way to Bar Harbour or a shorter way to Rockland. I'd go to Rockland and stay the night. Next morning leave your car on the mainland and take the ferry to Vinalhaven where you'll have booked a cabin for the next 3 days. If you choose the Bar Harbour option you can do Mt Desert Island and rent a place somewhere up there.
Day 5 - 7: Swimming, kayaking, lobster eating, reading, hiking, relaxing on Vinalhaven (or near Bar Harbour). There is absolutely nothing to do or see so when the day trippers go home it's blissful.
Day 8: ferry back to the mainland. Drive south to Portland, stop off in Wiscasset at Sprague's for lobster rolls. Don't bother with Red's, it's no better and you'll wait for 2 hours in line.
Day 9: spend the day in Portland, or morning in Portland, drive 2 hours south to Salem MA, visit the Peabody Essex Museum
Day 10: drive south to Boston, stay for a few days, visit Harvard and MIT, lots of walking for a change after all that driving
Day 14: drive to the Cape, stay a week with a day trip to Martha's Vineyard if you want, then return your car to Logan Airport and fly to NYC, connecting to your flight home. You can take the train but then you have to get out to the airport from Manhattan anyway so might as well fly.