I fly up there every year of two and love Maine. A great place for a long weekend is Acadia National Park. When I go, I fly into Bangor, rent a car from Budget (sign up for the email specials for the best deal) and drive to Bar Harbor, about an hour from the airport.
Quimby Inn in Bar Harbor has lovely Queen rooms for around $100, ask for AAA discount if you have it. Ask for a room in the main part of the house which gets locked at night, it's extra safe if you are travelling alone. You can walk from the hotel all over Bar Harbor, there is whale watching and shopping in town. Eat a lobster of course.
You'll need your car to see Acadia National Park. Oh, I think during high season in summer there is also a bus for around $20 that will take you around the park but I was there too early and had to drive. My favorite things to do are hike the carriage trails around Jordan Pond, and walk the trail along the cliffs from Sand Beach to Otter point, making many stops along the way just to sit on the large flat stone clifftops and watch the ocean.
If you really want peace and quiet and are willing to drive a little further, Lubec Maine, at the Easternmost point of the state (and US) is a wonderful place to get away from it all. It's a 3 hour drive from BGR. You can stay at a BB like Peacock or Home Port Inn, or get a room at the Inn at the Wharf with an ocean view. Take your passport and drive over the bridge to Campobello Island, Canada. The other really fun thing to do is drive the Quoddy Loop which takes you on 2 ferries to the Canadian mainland, Stop in St. Andrews for lunch and shopping,then drive back around to Maine. There's not much else to do in Lubec but walk along the beach - park at the water treatment facility and go when the tide is receding so you don't get stuck out there.
Both Bar Harbor and Lubec are tourist destinations so check online about kayaking and whale watching. Have a great trip, wherever you decide to go!