DS1 - 8/09
DS2 - 9/11
I live in Burlington VT. I would say that fun things to do with Children are Shelburne Farms (they have a petting farm and will let you milk cows, chase chickens, etc...) its alot of fun for kids.
Shelburne Museum is great for kids and adults. You walk around outdoors (so go on a nice day). There is an old-fashioned carousel, a steam boat, lots of buildings to visit and things to learn!
ECHO Center at the waterfront in Burlington. You can go here on a rainy day since its indoors. There are turtles to look at, frogs, snakes, starfish, hermit crabs, a horseshoe crab, lots of fish that are found in Lake Champlain. They have a room where the display changes monthly or semi monthly too.
Just walking up and down Church St in Burlington is fun for kids and adults.
Teddy Bear Factory is overrated I think. Plus the teddy bears are so pricey. Its an option though.
DD 11/2005
DS 4/2008
I agree with everything khalloc said above and definitely think you should check out the Stowe area as well. If you are in VT during the non-snow months you can drive up the Toll Road on Mt. Mansfield (the highest mountain in VT) and then hike the rest of the way to the top and then you could also drive through Smuggler's Notch. If the corn is high enough Percy's Corn Maze is supposed to be fun, and not all-day consuming like some mazes can be. If it's warm out, find a local and ask for directions to a swimming hole. Looking at the ages of your DC I think they would enjoy those things if they are at all nature inclined. In all seasons you can visit Ben and Jerry's, Cold Hollow Cider Mill, or take a hike. In winter, most places rent snowshoes, alpine and x-country skis for all ages. Also, we haven't been yet, but I've heard a lot of good things about the Pump House at Jay Peak and Montshire Museum of Science in Norwich, VT.