Japanese gardens - A place for philosophical contemplation
Japanese gardens are categorized into two types, one is landscape style and the other is Zen style. But there are few Zen-style gardens in Tokyo. The famous landscape gardens in Tokyo were built about several hundreds of years ago by feudal lords who were forced to come and go between Tokyo (called Edo at the time) and their home country every other year.
There are 5-6 excellent landscape gardens managed by the Tokyo metropolitan government that are open to the public in Tokyo. For example, Rikugien, Hama-rikyuu, or Koisikawa-Korakuen are very famous. Each garden has its best season by flower blossoms, so I will see the blooming season and your other destinations to choose the most convenient one and make an efficient itinerary.
If time allows, I recommend you visit Sankeien in Yokohama. It takes about one hour by train and bus from Tokyo but good to see. There is a huge and beautiful landscape garden with many old Japanese houses (some are designated as important cultural heritage) collected from all over Japan by the owner of the garden.
The last choice. If you want to see a Zen-style garden, by all means, you will have to go to Kamakura. It will take about 1.5 hours from Tokyo by train. As Kamakura is a very famous old city and the tourist destination, it sure is worth visiting. I know 3-4 Zen-style gardens we can see.