Tokyo typically sees blossoms in the last week of March/first week of April. It usually last for about one week. Rain can ruin it early. I think Hokkaido, the northernmost prefecture, is the season is early May.

As for places in Tokyo, my favorite are Aoyama Bochi (cemetary) -- people have midnight parites under these trees there -- and Chidorigafuchi area separated from the Imperial Palace by a moat. But lots of other places are there like Shinjuku Gyoen park (for a nomial admission fee), and Inogashira Park in Kichijoji, where some crazy drunken people jump into a pond there. People here usually use blossoms for an excuse to party.