April is a good time- it's warming up and the flowers are out, but it's not insanely crowded.
My favorite place for cherry blossoms is Dumbarton Oaks gardens. There's a fee, but it's something like $8 so pretty reasonable. They have a cherry hill which is lovely, and quite peaceful. The Tidal Basin WILL be packed. It's always packed.
Also, if you like Byzantine and Pre-Columbian art, the house has a great collection and it's free.
If you're interested in George Washington, Tudor Place down the street is a fun stop. It was built by Martha Washington's granddaughter and stayed in the family until 1983, when the last owner died and his will opened it to the public. There are several pieces owned by Washington in the collection. It also has a small but exquisite garden.