Shinjuku Garden would have to be one of my favourite places in Tokyo, and at 200JPY per adult ticket I would say it would have to be one of the most cost-effective outings, as well. Shinjuku Gyoen originated in the Edo period as the feudal Naito family's residence, and is now under the management of the Imperial Household Agency in Japan. It is a short walk from Shinjuku station and well worth the trip.
When I visited it was a couple of weeks past the main cherry blossom bloom, but there was still plenty of cherry blossom magic to be found as far as the eye can see. It is admittedly hard to take a photo of the blossoms without capturing at least five other people in the photo doing the same thing. The garden has more than 20,000 trees and over 1,500 of those are cherry trees.
Maps are available at the gate in both Japanese and English, and these are a necessity - the gardens really are huge! As well as the main garden, there is a formal French and an English landscape garden, although I did not find these anywhere near as impressive as the rest of the gardens. The gardens also feature a greenhouse with a plethora of different plants - I was particularly interested to see coffee and vanilla plants.
Many people take picnics along to the garden, but if you don't feel that way inclined there is also a restaurant in the grounds. There is a vending machine to order from and it is child-friendly with high chairs available. The staff were lovely to my family, and even gifted my son a little origami frog and a spinning toy. They also clapped their hands and looked very excited for me when I (poorly) attempted to speak some Japanese. The food was delicious as well (as was the sakura tea), and a great way to wind down after several hours exploring the grounds.