Without knowing details like how much luggage you'll have and where you're staying overnight, it's a little tricky to say. Personally, I would take Metro, as some have advocated. It's the easiest and quickest way off the airport property and into downtown D.C. Now, if you have significant luggage and nowhere to store it, I would probably opt for a rental car, though that just adds time and money and some complication to the equation.

I think it would help to know what you want to accomplish, how much luggage you'll have, and where you're spending the night. Also, when are you doing all this? Some museums (maybe all of them) have extended hours in the summer. For instance, the Air & Space Museum is open from 10 am to 6:30 pm from 5/27 to 9/5. If you land at 5 pm and have little or no luggage, you might stand a chance to get 30-45 minutes there, for example.

Obviously just seeing the monuments can be done at all hours, though I believe Park Police shoo people away in the overnight hours. As far as seeing the Washington Monument, the Lincoln Memorial, the Jefferson Memorial, the Vietnam Veterans' Memorial, and the Capitol, you can do all of that on foot, though probably not all of them "up close and personal" unless you're really committed to walking. You can see them by car, too, but only in "drive by" fashion, as parking and getting out and walking will be prohibitive, other than later in the evening.