Japan tourism ideas

• I would go then with a few days in Tokyo, just to get familiar with Japan, shake off jet lag a bit and get comfortable. I would definitely visit Shinjuku Gyoen National Garden and Meiji Jingu Shrine. The first one is a park and the second one is shrine but it is located in a very beautiful park.
• From Tokyo you can take day trips on a train to Nikko (hot springs, waterfalls, lakes), Hakone (lakes, volcanoes, hot springs, pirate boats and art gallery on fresh air), Kamakura + Enoshima Island (Great Buddha, temples, hiking trails, ocean)
• From Tokyo you can to to Mt. Fuji, but it has limited hiking period and requires permit nowadays if you want to hike it. Otherwise, you can spend some time around Lake Kawaguchiko, however, in my opinion it became too touristy recently, so unless you want to hike Mt. Fuji, I would skip it.
• From Mt. Fuji I would go to Kyoto (or you can go directly on Shinkansen there form Tokyo of you are skipping Mt. Fuji). I would leave several days on exploring Kyoto and visiting different temples. I especially like the Arashiyama area with its bamboo forest and a string of temples north of it. There is also a monkey park in Arashiyama.
• From Kyoto you can also do multiple day trips to Nara (to feed deers), Kurama (awesome hike with temples and hot spring at the end). If you are into matcha visiting Uji might be interesting.