Certain things require tickets. For example the butterfly pavilion in the Smithsonian National Museum of Natural History requires a ticket. You can get into the exhibit for free on Tuesdays, but arrive early to secure a ticket. They are for specific entry times. There is a cost on all other days
As noted, tickets needed for the Washington Monument.
Tickets are not required to visit the National Archives Museum to see the Declaration of Independence and the Constitution, but one can buy a reserved time for $1.50 each on line ahead of time to avoid the lines. This is worth while if one is visiting during busy times. Just about walk right in instead of an hour wait.
Haven't been, but my niece was really enthralled by the Spy Museum.
Others have noted the convenience of the Metro. Definitely the way to go (except to visit the Air and Space Museum out at the airport - a definite must in my book). Another option to get around is the
DC Circulator. We didn't use it as we needed the exercise, but it is convenient to cover larger distances once you're in town. Uses a SmarTrip card like is used on the Metro.
Do plan on doing some other things while in the area. We got "museumed-out" after five or six days, even though we broke it up with walks around the mall and part of a day out at the zoo.