It's better to go further from Tokyo. In Nerima ward there's one big park, Hikaraigaoka. It's big but doesn't have as many people as parks in the city and the blossoms are beautiful.
Shinjuku Gyoen. You can play sports there. There's a really nice lake and the grass is really soft so you can sleep if you want and, of course, there's loads of sakura trees.
Yasukuni Shrine. It's really nice with a lot of trees and stalls. You can rent out a space with the stall holders. It's free, as long as you buy drinks and food from them.
It's around the playground of Ikarashi elementary school in Niigata. There are cherry blossoms all up the surrounding hills -- it's like a stadium, but instead of seats there are sakura trees.
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Kunitachi is the best. I went yesterday. It's by Hitotsubashi University. The avenue is about 2 km, lined with cherry blossoms. There's a park in the university where you can sit. It's on the Chuo line.
Yokosuka Naval base. This coming weekend, the base is open to the public. It's an annual festival. There's cheap American food and Japanese food, but no shrines.