Memorials:

Lincoln, WWII, Korea, Vietnam, Arlington National Cemetary/Tomb of the Unknown Soldier.  Washington Monument (if you have time).

Museums:

American History Museum, Holocaust Museum, National Gallery, National Archives, Spy Museum (really fun for kids, but note that the Spy Museum isn't free, like the Smithsonians).

Further out (if you have transportation):

Mount Vernon - 30 min from DC (traffic can get bad)

Udvar-Hazy Center (my favorite museum in the area) - 40 min - 1 hr, depending on traffic from DC

Annapolis - 1 hour from DC

Harper's Ferry - 1:15 from DC

Gettysburg - 1:30 from DC

Monticello - 2:25 from DC