There are some nice places to stay in Conway and North Conway that are not too bad price wise. A place I stay at now and then is the Green Granite Inn on Rt 16 in No Conway. You should definitely take a ride down the Kangamangus (Rt 112) off Rt 16. It is a great ride on two wheels.
The ride up the Auto Road on Mt Washington is spectacular, but as someone else said, the edge of the road drops right off. The temp can be 80 deg when you leave the base and be below freezing when you get to the top, so have some extra clothing.
The Cog railway goes up the side of Mt Washington. It is a little pricey, but the view is unbelievable. I live about 3 hrs from Mt washington, so I go on several rides to that area every year. It is worth the trip. Enjoy!
Barkeep1956