I live in NH, grew up in White Mountains. The flume and basin are great. Hike Artists' Bluff right at the base of Cannon, that's a great view for little work. Take the tram up Cannon, some good views from up there. The cog railway is ok, but you can also drive up Washington for cheaper and more flexibility.
Just south of Franconia, is Lincoln, and the Kancamaugus highway. That's a great road, very scenic, lots of waterfalls and trails. I'd take that over to Conway and stop at all the spots along the way. Much to see on that road. It's spectacular in fall but good in the summer too.
Enjoy!
-Ed