Vermonter for the last 20 years here. You cannot succeed in seeing the best of Vermont without a car. Sure, you could take buses from some of the bigger towns into Burlington, but the countryside does not have the type of transport you are hoping for and you will miss out on some of the best things the VT rural and outdoor areas have to offer. Green Mountain National forest is pretty meh in my opinion as a landscape photographer/hiker. You'd have more bang-for-your buck up around Stowe, Groton Forest, or the Northeast Kingdom.
Why don't you want to rent a car? One of the comments here really touches on how much time you will truly waste if you attempt to do train trips or buses from all of these places. Vermont especially is tricky as such a large portion of it is rural.