I have been thinking about doing this too. I have been by myself but have never taken the kids.
Top of my list is a visit to Studio Ghibli! A friend told me the bus that picks you up from the station to get there is painted as a 'catbus'! My kids would have an absolute fit to be able to ride catbus - Totoro is one of their favourite ever movies. Anything to do with Totoro or Ponyo would keep my kids entertained for days.
Other kid friendly highlights:
Tokyo - Yoyogi Park on a Sunday - for the walk from Harajuku station past all of the Harajuku boys and girls dressed as manga and anime characters - or whatever else takes their fancy, the rockabilly guys dancing in the square, the hip-hop dudes, the 'battle of the bands' buskers playing around the outside of the park and the amazing use of space in the actual park. Every single bit of space was being used by people of all sorts. Sports, games, music, picnics… yet it still felt relaxed and spacious. It is a great example of urban green space being enjoyed by people.
Tokyo - toy stores - just cos they are there and they are huge!
Nara - Take the kids to squeeze through Buddah's nostril. With the deer park already mentioned, plus there is a giant golden Buddah in a shrine there that is impressive. In the shrine there is also a wooden column with a hole in the base that is said to be the same size as the giant Buddah's nostril. It is said that anyone who can squeeze through the hole will be granted enlightenment. I like the fact that only children can make it through and it makes them feel pretty special by the look on their faces. The temple also has some excellent scary looking guard statues.
Kyoto - random wandering through the back streets of the old part of Kyoto was wonderful. We found all sorts of interesting things going on off the regular tourist circuit.
Hiroshima - The shinkansen from Tokyo to Hiroshima was awesome! Bento boxes on the train, country life to watch through the windows and a fast fast amazing trip. The Peace memorial museum in Hiroshima was well worth the visit. It was difficult and moving but really well put together and a really important part of the visit to Japan for me. I am not sure how old your kids are and whether it would be appropriate for them age wise, but even just a walk around the gardens and outside of the museum and a visit to the dome would be good. The paper cranes everywhere are a great access point for kids to get a handle on some of the stories tied up in the bomb.
There is a beautiful island off the coast of Hiroshima - Miyajima. It is a short tram ride from the center of town and then a ferry ride to the island. It is a beautiful place with lots of deer, and one of the most impressive shrines I saw. It is the location of the famous giant Tori Gate - up there with the Opera House as an iconic vision of a country. When the tide is out you can walk up to the Tori Gate and touch it. There are also lots of walks and other activities on the island.
On the train back to Tokyo we stopped at Himeji castle. Lots of stairs to climb and an awesome view at the top. It is one of the oldest surviving castles in Japan and is a great glimpse into the past.
Something I didn't do but kids might enjoy - Sumo??
Ohh, I am so jealous!
Bookmarks