I was thinking of the last two weeks of Oct drsausage to include half term as I'd need to take DC out of school . From what I've read the best of the fall foliage is likely towards the beginning of the month in most areas (?).
OK - I mainly know Maine so can give advice about that. Maine is lovely in October, and things get a lot quieter after Columbus Day. I believe Acadia National Park visitor center is open till the end of October, and I really think October is the best time to visit - we go every year. Acadia is all about the hiking trails, beautiful mountains, kayaking in the ocean, etc. My kids loved Bubble Rock and Thunder Hole.
You should also see some lighthouses. The classic one is Portland Head Light, which is near some great beaches too.
The State Parks in Maine (many of them are beaches) are usually open till the end of Oct so you'll still have to pay to visit them, but the toilets and changing rooms will be open. My favourites are Crescent Beach, Kettle Cove, Two Lights, Popham Beach and Reid State Park. Popham Beach also has an old fort you can explore nearby.
The Botanical Gardens at Boothbay in Maine are magical for children - there won't be much blossoming in October, but they have a great children's garden, bear cave, fairy garden, meditation garden, lovely hikes, great cafe.
Later Oct you will probably miss peak fall foliage in northern New England, and it might be better to go further south if that's what you want to see.
The weather is more unpredictable than in summer, but it can still be absolutely beautiful - we do things like apple picking in October, go for lovely hikes (no bugs that time of year), and actually the ocean is sometimes warmish in October because it's warmed up all summer.
Boston is much more bearable in October than in the heat of summer - and as a PP said don't try to walk the whole Freedom Trail! When my kids were 8 they adored the Science Museum there (I still do). The Aquarium also is very good, and it'll be much quieter in October than in summer. A boat trip is fun, although check the weather forecast first - I did a quite rough whale watching trip from Boston and half the passengers threw up. My young niece and nephew loved the harbour tour, weren't too impressed by the Tea Party ship, did like the splash pond in the park.
You should read a few Robert McCloskey books before you come out - Make way for ducklings, Blueberries for Sal, One Morning in Maine.
We bought a National Park pass this year which was $80 and let one car in to all NPs. You can also buy State Park passes - we buy the Maine one every year, but it costs more and is probably not worth it for you.
Agree with others about it being expensive to eat out a lot. I'd stay in AirBNBs, cabins, etc if possible, so you have a kitchen. There is nothing quite so wonderful as a cabin by a Maine/New Hampshire/Vermont lake for a week.