For the White House, you do have to set it up WELL in advance with your Congressperson. For the Capitol there is a public tour, but you can also get a more "up-close" tour through your Congressperson's office; usually this is a private tour with a congressional aid. We really enjoyed doing this.
Monuments are free of charge, as are the Smithsonian museums. Last I knew, you needed tickets in advance if you want to go to the new African American museum; we got our tickets day-of but couldn't gain entrance until 2pm, which meant we didn't see the entire museum.
We are very glad we went to the Holocaust Museum, but you should know that it's intense, and maybe not a good choice for younger kids. Tickets for the Holocaust Museum are timed-entry, but that's only for the main exhibit. You can visit the side exhibits at any time.
Check out the DC Circulator. It's a bus system that is free of charge, with several loops in the city. The red loop goes by and/or stops at many of the monuments and museums.
https://www.dccirculator.com/getting-around-dc-faq/
Also, the subway is pretty easy to use/understand.
We have stayed out near Dupont Circle on our two most recent trips to DC, staying at airbnb locations.