The National Cherry Blossom Festival in Washington, DC is later this month: March 20-April 27, 2013. There are a lot of events around DC, e.g., special exhibits, screenings, etc. – some of which have already started! Many events are free so check the website for details.
The National Cherry Blossom Festival website states that the mean date of blooming is April 4. That of course depends heavily on weather and other conditions. However, if I were to plan a trip to DC, I would plan it around then and/or the National Cherry Blossom Festival Parade. You can check the blooming status on the National Cherry Blossom Festival’s “Bloom Watch” site.
My favorite location for the cherry blossoms is the Tidal Basin (the area where the Jefferson Memorial is). If you can go paddle boating, I highly recommend that. I also recommend bringing a picnic and eating under one of the trees.
Cherry blossom season in Washington, DC is a MUST DO.