Maybe it is not the best itinerary to start a Japan trip with one of its absolute highlights, Ancient Kyoto, and end it with the Mozu-Furuichi Tumuli. And I usually prefer to plan my trips with a top destination in the end. On the other hand, Kyoto was a perfect introduction to Japanese culture, architecture and food.
In September 2019, I spent five days in Kyoto, including a half day trip to Himeji-jo. I managed to visit 12 and a half of the 17 inscribed locations, I only left out the three northernmost sites – Kamigamo Shrine, Enryakuji, Kozanji – and Saiho-ji (which requires pre-booking). At first, I was a little worried about the busy schedule - three temples a day - and was prepared to skip a couple of sites, if necessary. Quite the contrary, not a bit of “getting templed-out”, I really enjoyed exploring all of these temples and shrines.

First, a few comments on the individual sites (in the order of my visits):

Kinkaku-ji or Golden Pavilion: the view of the pavilion with the pond in the foreground is iconic, and probably every visitor to Kyoto has taken this photo. However, there is not much else to see, and the interior was closed to the public, so all in all a little overrated.
The Zen temple Ryoan-ji was one of my favorites, it is most famous for the minimalist rock garden.
Ninnan-ji was a pleasant surprise. I was most impressed by the huge Nio-mon gate with its guardian sculptures, and by the Goten palace, the former residence of the head priest.

Shimogamo-jinja is a nice example of a Shinto shrine with its vermilion-coloured torii and buildings. It is located in an old forest, a perfect morning walk for early birds: the grounds open at 5.30 am (summer season), only the main hall opens later.
Ginkaku-ji or the Silver Pavilion was built as a retirement villa for a shogun. It was my favorite in Kyoto, mainly because of the beautiful garden. I arrived after a heavy rain shower, so it was much less crowded than I had expected.
Kiyomizu-dera is located on a hillside in the traditional Gion district, the most popular tourist spot in Kyoto, and is very crowded. The entire main hall was under scaffolding when I visited, but the view over Kyoto from the large veranda was not obstructed.

Byōdō-in (photo) in the city of Uji, south of Kyoto. The elegantly designed Phoenix Hall is depicted on the 10-Yen coin, but would definitely deserve a more dignified representation. Noteworthy are the Fenghuang (Chinese phoenix) statues on the ridge. The adjacent museum should not be missed and the modern Tonka Tea Room is perfect for a break.
Ujigami Shrine is just a short walk from Byōdō-in to the opposite bank of the Uji river. It has the smallest area of all sublocations. It was included because the main hall is the oldest existing Shinto shrine building in Japan.
Daigo-ji is my “half visited” site. I arrived late in the afternoon, too late for the one-hour hike to the upper part of the temple. But the lower part was also worth it. It is a large complex and - untypical for Kyoto – there were only a handful of visitors.

Nijo-jo, the former residence of the Shōguns in Kyoto. The watchtower in the south-eastern corner is popular photo motif, but the highlight is the Ninomaru Palace, especially the paintings on the walls and sliding doors, all in the style of the Kanō school. Unfortunately, it is not allowed to take photos inside.
Nishi Hongwan-ji mainly consists of two large halls and a connecting corridor. It is near the Kyoto station and easy to reach, but I found it less interesting compared to the other sites.
Tenryu-ji, another Zen temple with a beautiful garden. Most visitors come here for the famous bamboo forest, just next to the temple. Not bad, but a little overrated in my opinion.

To-ji, in walking distance to Kyoto station, most remarkable is the five-storied pagoda, the tallest wooden tower in Japan.

In summary, my favourites were Ryoan-ji, Ginkaku-ji, and Byōdō-in. The least rewarding were Ujigami Shrine and Nishi Hongwan-ji, but none of them were really disappointing.

A nice temple souvenir is a Goshuinchō, a book where you can collect the unique seals/stamps of a temple or shrine, not only in Kyoto but all over Japan. A book costs around 1200 Yen, a stamp usually 300 Yen. I have an almost complete collection of the WH temples and shrines I visited in Japan, only Nishi Hongwan-ji did not offer one.

Getting around in Kyoto
I explored Kyoto by bike, on foot, and by public transport.
I stayed in a ryokan about one kilometer north of Nijo Castle. My host offered me a bike and I used it for the first two days. Cycling in Kyoto is a bit messy, no bike paths, you can use either the road or the sidewalk. But it was a time-saving and relaxing way to visit the sites in northern Kyoto.
The sites in the southeast (Byōdō-in, Ujigami) and northwest (Tenryu-ji) are located along JR lines that are included in the Japan Rail Pass. Public transport works perfectly like everywhere in Japan. However, Google Maps was not very useful, so better use Arukumachi planner for Kyoto.

Non-WHS recommendations
As other reviewers already stated, Kyoto is full of temples, some of which are on par with the sites listed. Two recommendations: The Fushimi Inari-taisha is very popular, and rightly so; you should arrive as early as possible. The hike to the summit and back takes 1-2 hours, depending on how many photos you take. And second, Sanjūsangen-dō with the incredible number of 1001 life-size statues of the Thousand-armed Kannon.

Food is great in Japan, and Kyoto is considered the home of traditional Japanese cuisine (inscribed on the intangible cultural heritage list). If you're adventurous, the Nishiki Market is the place to go. I would recommend Tako Tamago, a small octopus filled with a quail egg. Not far from Nishiki is the Teo Kafon café, my recommendation is the Matcha Sweets Plate.

All in all, Ancient Kyoto is a great WHS and I would love to come back to visit the remaining four and a half locations.