A few months ago we went to Takayama (Gifu Prefecture 岐阜県) for a day trip (aka a couple-hours-worth-of-trip, if you include travel time), which was unfortunately not quite long enough to explore all we could. As you can see at their website, they have a lot to offer, from morning markets, to a squirrel park, to museums and shrines.
Unfortunately, as we could see from the map given to us, most of the places were were interested in (squirrel park, insect museum, Hida Temple, Teddy Bear village, etc) had entrance fees. So, most of us being poor and/or けち(cheapskates) we opted to walk up and down the narrow streets and check out the shops and free sights. (Oh, regret. Damn you, mind-obsessed-with-spending-a-cent!)
I recommend a trip here if you have the whole day. Bring some money for all of the amazing parks/museums/shrines you can visit. This is on my "I'd like to go back and explore more thoroughly" list.
One of the places near Takayama which had an entrance fee but which I did visit was Hida Village, a small village preserved in the traditional style, complete with artisans (such as carvers, whom you can watch work).
You can feed the fish in the pond behind.
Fish food 100 yen.
They offer kimono, props, and umbrellas to pose with in front of
the pond, all for free!
Homes and work places are preserved.
You are able to visit and enter each of these houses;
wear easily removable shoes!
Hida Village was very interesting, and also peaceful. It doesn't seem to take up much space but there is a lot to see. Another place I wish I had been able to explore more. Just outside the village there are omiyage shops, if you didn't buy something from inside (the items inside, being handmade, are a bit expensive).