A car is essential for anything outside the immediate downtown. The region is spread out in the way that rural Maine tends to be: Borestone Mountain is ten miles north on unpaved roads, Greenville and Moosehead Lake are another 35 miles beyond that, and Baxter State Park is roughly 90 minutes away. Downtown Dover-Foxcroft itself is small enough to walk between its two village centres without any difficulty, but you will not get far without wheels.
There is no public transport of any kind. If you are travelling as a passenger, plan on being completely reliant on whoever is driving.