Yes, you can go on public transport between Woodstock, NY (or close to it) up to Bangor, ME. You even have multiple options.
As found in the comments, the closest train station to Woodstock, NY is Rhinecliff, NY. Searching (in two separate times, since Amtrak's search engine does not seem to aggregate itineraries properly when there are too many connections), you can take the Empire Service down to New York (1h40) and then go by train from New York to Portland, ME with the Northeast Regional and the Downeaster. Portland to Bangor is operated by bus. It's taking pretty much an entire day (11AM to 10.30PM for example).
Greyhound also offers (cheaper) bus trips between Woodstock, NY and Bangor, ME. Once again, you will switch 3 times, including New York and Boston. The trip takes also a day (7AM to 10PM). The convenience is that you actually start from a bus station in Woodstock, NY (you don't have to have someone driving you 20 miles).
Rome2Rio shows you can mix bus and train, still switching (only twice !) in New York and Boston. The trip would also be long, taking you about 12 hours. And once again, the bus station is in Bethel Woods, NY, a 15 min walk from Woodstock (according to Rome2Rio).
So you have multiple options to travel with public transport in the Northeast. If you are interested in following your itinerary to Canada, there is a question covering the topic.