Every American shoudl visit DC. Not just to see how this fine country operates, but also to see the history and the arts and the culture that is abundent on the city on the Potomac.
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A trip to the Capital building is absolutely necessary, its a very cool building. Tours go around all the time. I haven't been to the White House, but a tour of that is also a must.
The tall obelisk that is the Washington Monument is the tallest building in the city. Though the line can be long to get in it is well worth it. The views from the top are amazing. The Lincoln and Jefferson memorials are also must sees.
A nice stroll down the mall is also key. Its a great area. The many museums of the Smithsonian must also be visited.
Sure, there are some bad parts to DC, but there are in every city. Other bad thing is the traffic, which DC is notorious for. Stay away from the Beltway, the highway that goes around the city in MD and VA... its one scary road!