When we lived out there, our first weekend we did the entire mall.
Took the train to Union Station, then walked to the mall. Made our way to the war memorials and saw Washington and Lincoln. Lunch from a street vendor. Spent the afternoon at the Smithsonian for American History.
8+ mile day with 4 kids from 10 down to newborn. Glutton for punishment?
How many days you have?
The "can't miss" is the war memorials, Washington, Lincoln, at least one Smithsonian, and the changing of the guard at the Tomb of the Unknown.
Vietnam Memorial, had the kids call my mom and ask her for a name of someone she knew. She graduated highschool in 71 so she had a lot of classmates that were shipped off to Vietnam. In this case, Richard Young.

If you have more time:
Try and get a ticket to go up the Washington Memorial. I think the view is pretty cool.

While doing the Mall, had my 4th grader do the Junior Ranger program. Kept her busy and focused despite the miles.

FDR and MLK aren't on the mall, but over by Jefferson. You can walk there, but best if planned for a different day than the mall.


Smithsonians:
When I did the American Indian one in 2006, I wasn't too impressed.
Kids loved Air and Space. Remember there are two. One on the mall and one out by Dulles. The one at Dulles has the space shuttle.

Apollo 11 on the mall

Discovery at Dulles
American History was pretty cool, mainly for nostalgia. My almost 3 year old saw the Count and immediately started counting.

Natural History didn't have the dinosaurs, but they are back. Half of it is done extremely well, you almost feel like you are walking in the African savannah, the other half, well, maybe it is better now, but it was just a museum.

We toured the capital once. Contacted Ted Cruz's office and got a tour plus tickets to the gallery of the house. When I went in 2008 during the Republican protect, Representative Ted Poe took us to sit on the floor of the house to listen to the Republicans lecture the not there democrats on drilling. If you have the time, its free, just contact your senator or representative. Ted Cruz's staff gave the kids dr pepper, so they were pretty happy from the get go.
White House tours I hear have gotten better. The two I've taken really weren't worth it. Saw a couple of rooms and had to store your phone outside the building. Allegedly you can take cameras now.
Supreme Court we never toured, but you can do a "quick tour" if the court is in session and sit in the gallery for a few minutes.
Arlington, the changing of the guard is a must. Even my kids were impressed. Going to the Iwo Jima memorial is cool, famous statue. My kids didn't fully understand the importance as two of their great-granddads were corpsmen who served with the Marines, but still went. I try at the memorials to make it more than just a picture spot, but talk to them about those in our family who have served and talk about the cost of freedom.
Will Spring Break line up with the Cherry Festival? Never made it, my mom came out for it but we had a heavy freeze a few days before that canceled it. Cherry blooms frozen in ice isn't that pretty. But the street we lived on was lined with Cherry trees so a bunch of cherry trees in bloom is pretty.
Library of Congress can have a wait, but you get to see the constitution and the declaration of independence. Plus they have a Guttenberg Bible on display.
On another trip we did the museum of the Bible. Kids surprisingly enjoyed it. Depends on your faith tradition I guess.
Georgetown Cupcakes. If you have the time, and the family watched the show, could be worth a stop. We combined it with a walk around of Georgetown University.
The pentagon, went there for work so don't see the need to be a tourist, but if you aren't military, may be worth considering.
My favorite hidden place? Theodore Roosevelt Island.