The DC Cherry Blossoms are absolutely gorgeous. The "peak bloom" date will shift year to year but this should be around the time to catch some beautiful flowers. It can be very crowded during peak bloom so parking will be difficult. We recommend that you park further away and metro to the Smithsonian stop (Blue/Orange/Silver lines).
All the Smithsonian Museums are fascinating in their own way. A Shannon family favorite is the Air and Space Museum. All of the Smithsonians are free to enter and are generally within walking distance of each other. There are parking garages around the National Mall that you can use.