My husband and I want to take a road trip from Asheville NC to Vermont this October. What. Places do we need to stop along the way or make sure to see in Vermont.

I would go see Shelburne, an open air museum near Burlington.

What are your interests? How long do you have for this trip?

Ashville to Burlington is a 15 to 16 hour drive non stop. Assuming this is to see foliage, you need to aim for northern most destination where color starts first. Many places will require a two night stay and October reservations go quickly.
Shelburne is a great suggestion, Burlington is a interesting city. Maybe someone can remember the name of the ice cream/deli place with an outdoor deck that looks out on Champlain. Get there early since it's a popular place to watch sunset.

The west side of the state has open farmland with rolling hills as you near Burlington. The east side of the state is more hilly so a different look at foliage.

Visit a farmers market on a saturday. The one in Norwich is great. So is the one in Brattleboro. If you like to bake you might be able to sign up for a baking class at King Arthur Flour in Norwich.

What are your interest? How long do you have for the drive? Is this a round-trip, returning to North Carolina?

How long will you be in Vermont? Are you trying to time it to see the Fall colors ("leaf peepers" haha)? I used to live in Brattleboro VT (New Hampshire and western Mass). Brattleboro is a fun town at the SE corner of the state (nearilsh the border to New Hampshire to the east and Massachusetts to the south). Western Mass around Amherst and Northampton is worth a stop (as you are nearing Vermont).

Drive the Kancamagus Highway in New Hampshire on your way to Vermont, it’s one of the most stunning fall foliage routes in the U.S.!

Depends the route, coming from the south a person wouldn't necessarily even be in New Hampshire in order to get to Vermont