I'd stay on the VA side, just so you'd have some evening options for dinner/etc (plus, I prefer VA over DC at night just for safety - Arlington and Fairfax Counties have some of the lowest crime rates anywhere). I'd also stay by the Metro (within walking distance or by free hotel shuttle).
As for activities, it depends what you like. There are many free sites, many free sites that require reservations, many pay sites, many pay sites that require reservations. Also, you will be in DC during the Cherry Blossom festival, so there are those activities, too.
If it were me, I would plan one day for a tour of the Capitol (
https://www.visitthecapitol.gov/plan-visit/book-tour-capitol), including possibly seeing it in session, and a tour of the Supreme Court (
https://www.supremecourt.gov/visiting/touringthebuilding.aspx). I'd plan one day for a tour of the Smithsonians - favorites are Natural History, American History, and Air and Space, but all of them are pretty awesome - you might want two days for this item if you love museums. Then, I'd plan a day enjoying the cherry blossoms and pay sites - maybe a day at the Newseum, the Spy Museum, a boat ride (or kayak) on the Potomac, Madame Tussaud's, etc. And I might spend a day in VA at either the Pentagon (again, they do tours - sign up ahead), Mount Vernon, Arlington National Cemetery, Air and Space in Dulles (which is better than the DC one if you have to pick), etc. Of course, there's still 100 of other free sites (Bureau of Engraving and Printing, the Archives, the National Arboretum, the White House, etc), but that would already fill up the 4 days pretty fully...
I wouldn't pick more than 1-2 pay tourist things, b/c there are so many nice free options everywhere (but the pay options ARE worth it - you just don't have enough time to even see all the free stuff
...and I would not continually criss-cross the city - do one area for a day and save the next area for the next...like the Capitol suggests if you have a day...
"Start with the Capitol tour, visit Exhibition Hall, watch Congress in session, and then take some time to explore the offerings of our Capitol Hill neighbors, the Library of Congress, the U.S. Supreme Court, and the U.S. Botanic Garden."
Last piece of advice - I wouldn't do all the same thing. So, I wouldn't spend 4 days in museums, or 4 days touring government offices, etc. I'd want to have a little of everything, indoors AND outdoors, since it should be really nice during late April.