Photo: Fabimaru, CC BY-SA 4.0 via Wikimedia Commons
In autumn, the beautiful orange and yellow tones of changing leaves ("momiji") are beautiful, and there is also a nice view on the lake and the mount Fuji.
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河口湖 Kawaguchiko
The easiest to reach and the most touristic of the five; it's also the only one to have an island. There are several views on the Fujisan around the lake. You can see its reflections in the water from the north shores surrounded by cherry blossoms (spring) or lavender field (summer). You can also see the remains of the old lava flows from the Fuji to several places around the lake (in particular on the south shore)
もみじ回廊, Momiji kairō
Another celebrated tree-viewing spot. A variety of beautiful trees, including maples and gingkos, line a small river on a gentle slope. The trees are lit up at night during the autumn tree viewing season.
河口湖木ノ花美術館
A museum of the popular Japanese children's book "Dayan the Cat" by Ikeda Akiko. Shows a lot of artwork from the artist's books in the world of Wachiworld. Has a lovely giftstore and cafe.
富士山パノラマロープウェイ Fujisan panorama rōpuwei
This short cable car climbs to a viewing platform next to mount Tenjō. There you can enjoy a nice view on the lake below and on the mount Fuji, and also have a tea and eat dango (rice flour dumpling).
三ツ峠山
The hiking time is 3 hours there and 3 hours back. The path is fairly well maintained.
青木ヶ原
Also known as Jukai (樹海) or the "Sea of Trees", this forest is famous for its volcanic caverns and infamous as Japan's top suicide spot — in an average year, over 70 corpses are found.