I spend a lot of time in Vermont, not as much in Burlington. Highly recommend visiting Stowe if you have a car and can get out. At that time of year with the foliage, you could do a nice drive. The Smugglers notch area would be particularly nice. Cold hollow cider mill is fun. The Ben and Jerry's HQ is there and you can do a factory tour. We always end up getting breakfast at the Butler's pantry in Stowe. They have a bike path that goes through the whole valley and plenty of bike rentals.

Burlington is a fun town with a cool vibe. I'll let others give recommendations on best things to see and do there. Great time of year to be visiting.