Squirrel forest Hida wild flower gardens
(Risu no mori // Hida-sanyasou-sizen teien
20min by car from station
2351-7 Matsukura-machi
I hope this helps.
Waterfalls....there are some. I think it is called 101 or maybe 1001 waterfall. Something like that. Sorry i can't be precise. It is a nice walk in the nature, a bit uphill though.
A bit further from Takayama you have the Hirayu Otaki - a large waterfall about 6 meters wide and 64 meters high. Just ask for Hirayu onsen.
There is a playground in the Ninomaru park which is a part of Shiroyama park. Locals go there with their children for a picnic. You can see most of the Takayama from there, because it is on a hill. Nice, clean and quiet. Just ask for Shiroyama park or Higashiyama walking course.
There was also a tiny petting zoo...well actually it is an old lady with her hens and ducks and a cow. She also has some aquariums with koi fishes and some other big fishes. We were warned not to touch cuz they could bite of our finger ;)
But i can't remember where exactly it was. I'm sorry. I know we were on a daytrip to Hida Nyukawa Cosmos Festival, which is on a mountain just above Takayama, and we stopped at this old ladies petting zoo. It was really nice. It is an old wooden house and the entrance fee was really cheap. Please try and describe it at the tourist office. Maybe you are lucky enough and they know where it is. Because you will have a car, it would be pity not to visit all the tiny villages around Takayama. You will feel like you are not in Kansas any more ;)
If the tourist office is of no help, then you could try and ask at the J-Hoppers Hostel Takayama. They do adventure trips.
http://takayama.j-hoppers.com/e_accessmap.htmlPlease enjoy Hida Takayama, and if you have any more questions, just ask. You can also look for HidaTakayama on facebook and join.
Enjoy!