Where in New Hampshire are you? If you're just looking for day trips, it might matter if you are north or south in NH. I lived in Vermont for 30 years, now live on the west side of Lake Champlain and look out my window at Vermont every day.
If you come from the northern regions, then I'd recommend the lake region/ Northeast Kingdom area. Lots of beauty up there. St. Johnsbury is worth a stop, Fairbanks museum is impressive. Jay Peak is now a year round resort but I can't speak to their covid restrictions.
If you're crossing in the Lebanon/Dartmouth region then I'd suggest a ride over to Bethel Vt and a gap ride to Rochester/Hancock and then another gap to Middlebury along Rt 125.Stop at The Rochester Cafe for lunch or breakfast, you will not regret it. Just a spectacular ride through the Green Mountain National Forest and quintessential small town New England. Middlebury is 30 miles south of Burlington and the aforementioned, and highly recommended Shelburne Museum is about 25 miles north.
Crossing down south I'm less familiar with, but I'd guess that you can't go wrong by motoring over to Rt. 100 and heading North. Woodstock, Quechee Gorge/Simon Pierce are all along that route.