I have spent a great deal of time in Kyoto and it never ceases to break my heart.
The city itself is an utter monstrosity, a nightmare mess of urban planning.
There are some wonderful spots in the city, but these are marred by the city itself. Many of the temples, gardens and shrines are almost ruined by tacky souvenir stalls that seem to dog this country's tourist sites.
I have visited virtually all of the temples and sites inscribed on the list and would recommend the following:- (1)Byodo-In, one of the few remaing Heian period buildings left in Japan. It is a real gem. (2) Enryaku-Ji. A wonderful mountain-top temple that has beautiful views over Lake Biwa. (3) Nijo-Jo. One of the most outstanding palaces remaining in Japan.
The most visited temple is Kinkaku-Ji - the Gold Pavillion. I find this place rather gaudy and overdone. It is a relatively recent creation, built in 1955 after the original was burnt down.
Nishi-Hongan-Ji is currently undergoing a massive face-lift and rebuilding project. The Kondo Hall will be off limits for several years.
Ginkaku-Ji (The Silver Pavillion) is looking a bit worse for wear these days and could use a lick of paint.
Re-building is something that always has to be born in mind with Japanese temples - what you are looking at may be advertised as being 700 years old, but it could easily be a modern re-construction.