There are plenty of places to go to in New Hampshire as well as Vermont. Have you considered NH's Lakes Region? That is on the edge of the White Mountains, and as the name suggests, there are numerous alpine lakes in the area, along with a lot of really nice small New England towns. There are lots of things to see and do, and the whole place is very family friendly. An alternative might be the NH seacoast area- that area too is also family friendly, and it's possible to drive from one end of NH's coast to the other in about 20 minutes- the coastline is only 18 miles long. There are lots of things to do here, too. From festivals like the Strawberry Banke in May to the Clam Feast in late summer, there is something for everyone here. The Lakes Region has Gunstock State Park, Endicott Rock State Beach and waterpark, Ellacoya State Beach, Mill Falls Inn and Marketplace in Meredith, ( which incidentally is a GREAT place to stay as well as shop and eat) and attractions like the Flume, Lost River, Canon Mountain ( home to the famous airial tramway) Castle in the Clouds, and steamboat cruises on Lake Winnepesaukee. A short drive to the east, and you can visit Conway and East Conway, and cross over the state line into Maine, if you want. These towns are famous for their association with one of the major tennis tournaments on the entire East coast- and they have played host to the US Open championships more than once.
Since I am not sure of your interests or budget, I hesitate to suggest actual activities, but I can tell you there are lots of nice places to stay in NH which are affordable. One of my favorites when I go to visit there ( I'm an NH native with strong family ties in that region) is the Landmark Hotel in Laconia. Although they are a limited service hotel ( they don't have a pool) they provide good service and free breakfast every morning, as well as reasonable rates. There are other resorts around which also offer good service, such as the Margate Resort, which is also owned and run by the same people who run the Landmark. Check them out- they are great places to be !!
I hope this helps you with your planning, and enjoy your trip !!