The 2 week period from the end of March to the beginning of April, when the cherry blossoms are blooming, is a very nice time to be in Japan. 🙂 I’ve been on a one week break from my Japanese studies (I passed all my tests and made it to the next level), so together with friends I’ve been going to Shinjuku Gyoen park, and climbing Mount Takao, among other things. If you want to look at cherry blossoms, Shinjuku Gyoen is a very nice park, with lots of nature and nice scenery, and I can recommend it. By comparison I would recommend staying away from Ueno Park, as it’s nowhere near as nice a place, is mostly concrete, is much smaller, and more importantly, when you get there you will most likely be in for a shock as to the sheer amount of people that are there. At its most hectic, it feels like you can barely move around in the main street with the cherry blossoms.
Mount Takao is a mere 1,5 hour train ride from Tokyo (that’s from Shinjuku). It’s only 599 meters tall, and has lots of nice scenery. It’s popular with hikers, but there aren’t too many people or anything. You can see Mount Fuji from the top as well (granted you can see Mount Fuji from a lot of places in Japan. It’s a gigantic mountain, so it can be seen from extremely far away). I highly recommend it. 🙂