Okay, so you're in Washington, and you want to do touristy things?
If you had any sense and wanted to save some money, you stayed slightly out of town and enjoyed reasonable rates for your accomodations.
If you're easily amused, and have good legs, try walking around town.
Washington is replete with museums, in fact you can live here for years and not see them all in their entirety. Besides, there are frequently special exhibitions for your pleasure.
If I was a person of the leisure class, who likes to walk, and if it were a beautiful sunny morning with reasonable humidity and temperatures, I believe that I would pack an umbrella anyway and then take the Metrorail downtown, and get off at the dimly-lit Union Station exit. (This is also a prime arrival gateway into the city via Amtrak.) From the lightless recesses of this basement platform, one emerges to confront a vista of lovely Capitol Hill, and of course the magnificent Capitol Building itself. Be careful not to gawk or you could be run-over by yuppie scum from hell as they scurry to their meetings and assignations. Proceed towards the Capitol and then gawk - I've lived here since 1963 and I never get tired of seeing that building. Wander around. Take pictures. Everyone else does. Then go back to Union Station and check out some of the shops. Many are pricey, but many are also priced to serve the rail traveller. It's a great place to stock up on bottled sodas and extra film.
Walk down Pennsylvania Avenue to the West, which is to your right as you look from Union Station towards the Capitol. Check out the architecture. I particularly like the Canadian Embassy, and the National Archives building. There's amazing statuary all over the place. This is the truly-grand part of Washington, and if you're afoot you can easily ignore all of the bus-swallowing potholes in the streets. Keep walking down to the Navy Memorial, where there's often a band playing. Look around the historic Old Post Office - you definitely don't see this sort of architecture every day. Hungry yet? There are many restaurants, ranging from fast-food and delis to fairly upscale eateries. If you've been walking slowly, this is a good time for lunch. If you didn't pack bag-lunches (recommended) you can actually rub elbows with harried Federal workers and assorted law-firm lackeys during the regular lunch-hour stampede.
After eating, wander up to 15th street, where you can see the back side of the White House. Wander around to your left as you approach, and enjoy the sight for a while. Then unless you feel some strange need to see the wedding-cake monstrosity of the Old Executive Office Building, head south, down to the Mall. Look back towards the Capitol, and then walk back towards it. There are enough museums there to keep you busy for a month. Unless you see something that particularly fascinates you, just wander around the outside of the museums - there are a great many exterior displays, and probably a great many tourists who might be as interesting as are the museums. But be sure to write down or remember which exhibits are at which museums, because you'll be coming back tomorrow.
Head back to your hotel or wherever, and get something to eat. Then stay in and watch the local news, and read today's Washington Post.
Wherever possible, ignore the locals. Mostly they want one or both of two things: your money, and for you to go away and take all of the other tourists with you.
Go back and look at all of the museums you wanted to see! that should take you at least all day - then go back to the homestead and get some rest.
Take the Metro Blue Line to the Arlington Cemetary exit in Northern Virginia. If you have a relative here, be sure to pay your respects. Then make your way carefully towards the Memorial Bridge across the Potomac River, and head across. There's quite a view to be had from this bridge. You'll be arriving at the Lincoln Memorial. Wander around, it's really very impressive. Then walk downstream along the Potomac, and across the parkway bridge over the Tidal Basin, and enjoy the vistas afforded from the Thomas Jefferson Memorial. Get your bearings on the White House, which is very easy to see, and then walk towards it past the Bureau of Engraving and Printing. when you get to the Mall, go see the Washington Monument. If you want to stand in line, take a ride up to the top and look around.
Now head west, back toward the Lincoln Memorial. Be sure to visit the Korean War Veteran's Memorial and the Vietnam Veteran's Memorial. Be prepared when you visit the last, as it often has quite an effect on people.
Take Virginia Avenue to the Northwest (any northbound street will intersect it), and dine and wine in fashionable and touristy Georgetown.
And on another day, or two, you might want to visit the lovely and cosmopolitan (and occasionally very weird) Adams-Morgan Neighborhood.
Or perhaps you'd like to drift through the Dupont Circle Neighborhood