Suggestions for Tokyo and Kyoto?

|The Volokh Conspiracy |

epa04771810 (01/19) Pedestrians walk along the Horikawa river near Modoribashi bridge in Kyoto, Japan, 17 April 2015. The area is believed to be a gateway between the human and spiritual realms and is located near Seimei shinto shrine, where the spirit of the 11th century court diviner and master of esoteric Japanese cosmology, named Abe no Seimei is enshrined. The 1,200 year old city of Kyoto has lately seen a renewal of ancient traditions and spiritual beliefs by a rather young group of citizens. They see their activities as the beginning of a new culture that may eventually find its way into the mainstream just as young people of the past once did. It is a world that few outsiders become privy to. EPA/EVERETT KENNEDY BROWN PLEASE REFER TO THIS ADVISORY NOTICE (epa04771809) FOR FULL PACKAGE TEXT

My family and I will be in Tokyo and Kyoto during the next couple of weeks; any suggestions for cool things to check out, beyond the obvious? Tips for tasty and unusual restaurants would be especially welcome. And of course any recommendations for me would also be recommendations for others who come across this post, now or later. Many thanks!