Being from Vermont I can tell you that there is practically no place in general that you can't carry for self protection. The parks notice referred to above simply says no horse play. No plinking, casual target shooting, and that kind of thing while the parks are open for the summer. Most likely hunting is allowed during off season.
There are some towns, Montpelier being a notable example, where they have tried to pass ordinances that firearms are not allowed in the city limits. (State Capitol) (Courts are off limits) Even we residents don't know from year to year what's up with those as any town can hold a meeting of their governing authority and pass ordinances which have to be challenged to be found unconstitutional. Our sportsman's clubs are good about challenging most of the crap that comes up on state and local levels.
You would probably be best served by contacting a fish and game club closest to the town you'll be visiting and asking for their most knowledgable person to help you. There is also the Vermont Federation of Sportsman's Clubs at
Vermont Federation of Sportsmen's Clubs Home Page that has some very knowledgable people. Whatever you do Vermonter's would appreciate your membership and or donation to a club or the Vermont Federation of Sportsman's Clubs to help us keep free carry available and the special interests off our backs. We're a small State losing hunters and firearms owners every year, and we're losing ground to the lead banning crowd, and "sensible" game protection crowd.
Annual dues are very reasonable as you'll see at most of the club websites.