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If staying in Alexandria try for some place economical in the Old Town area, although economical and Old Town don't usually go together. Again if staying in Alexandria, flying into Reagan would be the first choice, Dulles second. The Metro provides very good transportation around DC and the suburbs but some places of interest - Mount Vernon for example - may require a taxi or bus ride too.

If you want the White House tour, try working through your Congressman to arrange tickets. Well in advance! The cherry blossoms reach their peak in early April (typically) so you will miss that.

A new museum that has opened in the last couple of years is The Spy Museum - get tickets ahead of time if interested because it has been selling out. Even newer is the aviation wing of the Smithsonian which just opened in late December 2003. It is located adjacent to Dulles Airport and would require taxi (the Metro doesn't go out that far). Admission is FREE. Although I have not been yet I have heard from other visitors it is fantastic - lots of static displays of famous aircraft, from the Enola Gay to one of the shuttles.
That's the tip of the iceberg. Let me know if you need more info.....