This should be helpful: http://www.nynjtc.org/region/appalachian-trail?gclid=CJ-hgqeui7ICFYeDQgodqRsAWw
How much of the trail do you want to walk? There are people who walk a section of the trail and have someone meet them to take them back to the original vehicle.

I really just wanted to drive the scenic route and end up in Vermont somewhere, like on a lake i'd say.

One suggestion might be to take I84 over to Danbury, CT and then take Route 7 North through scenic areas all the way to Vermont. River, mts, small villages etc.

We like the Delaware Water Gap area. We go up to Albany and then cross over to Bennington VT. You can take the route over Hogback Mountain and end up in Brattleboro. If it's a Saturday, you can go to the farmers market which we thought was very festive and a great place to buy an ethnic lunch. If it's not saturday then you can get bahn mi sandwiches or noodle salads at the Humble Kitchen food truck parked in the Harmony lot in downtown Brattleboro.
Go over to Keene. Nice college town. Head to Concord NH. This will be a rural state hwy. You'll end up on Rt 202. You can browse a used bookstore in an old barn in Henniker (Book Depot). Turn left onto Rt 127 and head into downtown Contoocook. In the village turn right onto Rt 103. This is a drive thru some pretty villages. Make a slight detour to Gould Hill Orchard which Yankee magazine just announced as the orchard with the prettiest view in New England. They also have a wide selection of heirloom apples.
Backtrack and continue to downtown Concord. If you are feeling weak from hunger, Bread and Chocolate is a great European pastry shop on S Main St.
Get on I93 North. If you keep going you will end up in the middle of the White Mountains but you can take a side trip to Lake Winnipesaukee (exit 23). Maybe ride the MS Mt Washington. Take a drive along the north side. Head to Squam Lakes. Visit Center Sandwich.

If you have the time, there's a sweet like lake in Washington NH (state park) and Hillsboro Center is well worth the detour.

Instead of heading to Concord you can head over to Bradford NH and then up to North Sutton where there's a lovely little lake. Then head up toward New London (pretty college town) and maybe a drive up the Wilmot side of Mt Kearsarge. (toll road) for some splendid views. You can drive around Pleasant Lake in Elkins. These are much smaller and quiet than big Lake W.

Instead of heading to the White Mountains from New London you can get on I89N to Vermont.

There are lots of options.