Stephen King still owns 2 homes on West Broadway in Bangor, although he doesn't live there year round. His main house is the red one with the wrought iron fence around it (used to have bats on the fence posts but they aren't always "up" as people steal them) as well as a more recently-purchased home next door (to the right as you face the house). Last summer we took visitors to see his house and although it appeared empty, suddenly a buzzer rang and the driveway gate slowly swung open. Nobody was around, nobody drove out or into the driveway...just that slowly opening gate. Kinda creepy!
It's hard to recommend someplace to go to where there's lots around to drive to. Bar Harbor (about a 5+ hour drive north of Boston) is nice for a day or two; the town is quaint, there're lots of small shops in town, and you could spend a day driving around Acadia National Park and walking/hiking around on top of Cadillac Mountain. Not sure I'd stay a week, though. Bangor isn't particularly exciting, although there's the Cole Land Transportation Museum, a pretty decent Children's Museum, and a cute downtown area. You could also walk around on the campus at UMaine and go to the planetarium there (but check the schedule). If you are hikers, you could drive north to Baxter State Park and hike Mt. Katahdin, but you have to be there EARLY in the morning.
Portland is a nice small city. Great downtown area, the Old Port, lots of shops, restaurants, microbreweries. It's about 2 hours' drive south of Bangor. My family lives in York, on the southern seacoast (about a 90 min. drive north of Boston). If you found an airbnb in York, York Harbor, York Beach you could spend your days on the beach, drive up to Ogunquit for shopping, over to Kittery to the outlets. Portsmouth, NH is just south of York, where there's once again small town quaintness, shopping, great food, Strawbery Bank Living History Museum, outdoor evening plays or movies at Prescott Park, or a trip to Water Country if your kids prefer water parks to the ocean. In York Beach there's York's Wild Kingdom (haven't been in decades but friends say it's fun), and York Village has some historic buildings that you can tour. You can drive north to the Seashore Trolley Museum in Kennebunkport. K'bunkport is another quaint small town with shopping, food, etc., and you can do a drive-by of the George HW Bush summer home out on Walker's Point. You could even drive a bit further north (but not as far as Portland, which is about a 45 minute drive north of York on the highway) to Old Orchard Beach for a day on the beach and an afternoon/evening on the Boardwalk. IF you really want some history, you could drive about 90mins south of York and go to Salem, MA, or even go into Boston for a day... a very busy, full day!
Also... I don't know if it's your thing, but Maine is FULL of microbreweries; if you stay anyplace from Kittery to Orono, you could spend a day visiting several microbreweries in whatever vicinity you're staying in; in the very small college town of Orono, we have three microbreweries within walking distance of one another, all brewing very good, very individual styles of beer. I am sitting in a hotel room in Portsmouth NH right now, and can think of 6 microbreweries within a 30 minute drive from here.
Because I summered in York and now live in Orono, I don't know too much about the western part of the state, or the lakes area... but perhaps someone else can chime in?