If there is a special exhibit at one of the museums that attracts many you may need a ticket in advance - usually free but ticket services can add on a charge for the service. You can also try going there early in the morning to see if there are any tickets available for that day.
The White House requires a ticket - either by getting there early in the morning, getting on line and then getting a ticket for a time to come back that day. You can also contact one of your Representatives - Congress or Senate and arrange with them for tickets for a White House tour - same tour but you get the tickets in advance.
The Capitol may be doing the same thing requiring getting a ticket early in the morning with a tour time on it for later in the day.
Smithsonians are free and require no ticket. Monuments require no ticket, other than possibly the Washington Monument to go to the top if it is peak tourist season - again no charge but you have to get it early and go back later. There is no parking at the Jefferson Memorial any longer and it is easiest accessed by TourMobile - or if you park at the FDR Memorial and walk around to the Jefferson along the path.
Travel outside DC and you can go to Old Town Alexandria, Arlington, and also to Mount Vernon. Mount Vernon charges an admission. Go with enough time to see the museum and presentation center, tour the grounds, the farm - all in addition to the house. There is also a working Mill and Distillery down the road owned by Washington.
Lots of Civil War sites near enough to DC in Virginia - Manassas and Fredricksburg. NPS sites and there may be a charge for the museums but not the battlefields.