Sakura trees come in several different kinds, and the most famous one called “Somei-yoshino” is regarded as synonymous with Sakura. Though Somei-yoshino blossoms are so delicate that they last only 1 week or so, other kinds of Sakura like Yaezakura can be enjoyed much longer. Since Yaezakura begins to bloom around when Somei-yoshino ends blooming, you will be able to enjoy Sakura more than 3 weeks.
You don't have to go a famous place to enjoy Sakura viewing.
You will find your favourite Sakura viewing spot in a neighborhood. During the blooming season, Sakura can be enjoyed here and there in Japan, at parks, gardens, schoolyards, along streets and riversides. Viewing Sakura while strolling in towns will be fantastic.
Recently, famous Sakura viewing spots tend to be too crowded with tourists, but the blooms at those sites are still outstanding. To find famous Sakura spots in Tokyo, see this page Sakura Cherry Blossom Viewing in Tokyo. The information in this page is about viewing spots of Somei-yoshino.
Sakura Cherry Blossom Viewing in Tokyo
Somei-yoshino along a river (Kawagoe, Saitama)
Yaezakura as roadside trees (Tokyo)