Depends on what you're looking for. Regardless of where you go, I'd recommend going in September. The summer people have cleared out, mostly, and the leaf-lookers haven't arrived yet. Weather is generally good, or as good as can be expected in Maine.
If you want to be a tourist, then just land in Maine, start at Kittery (southernmost point on the coast) and work your way up Route 1 to Acadia National Park. Hit the Old Port in Portland, LL Bean in Freeport, and just about any "folksy coastal town" along the way. You'll see everything that everyone else talks about and will have a good time. You'll also deal with mobs of tourists, and pay through the nose for everything.
If the outdoors is your forte, Moosehead Lake + Baxter State Park (central northern part of the state) are great, and a little more off the beaten path. The Appalachain Trail starts in Baxter. If you REALLY want to get lost, the Allagash Wilderness Waterway (far northern part of the state) is a good week or two in the woods. I would only recommend this if you're pretty experienced, woods-wise, though.
If you want to tour around Maine, see some great stuff, and NOT be a tourist.... I recommend the Western part of the state (where my family is....) There's plenty of state parks, hiking, fishing, and gorgeous scenery. Not a whole lot of folks from away, particularly if you go in September. Rangeley and the surrounding area, in particular, is beautiful. Don't expect a lot of nightlife, restaurants, and clubs to go to though. The streets roll up pretty early. A wild night out may consist of sandwiches picked up at the local general store, eaten on a picnic bench overlooking the lake. Maybe icecream from the dairy bar, and a good book before going to bed.
Well said, & very helpful. Thanks. You represent Maine well.