I used to live in DC, (before there was the subway) but I know that these days, the public transportation is your easiest way of getting around. Do not drive in DC unless you absolutely have to. Perhaps things have improved, but the Capitol Beltway (I-395) was a PITA to negotiate.. Downtown, the streets are mind bogglingly insane (one-ways abound, the streets are set in circles in parts and in straights in others, traffic is bumper to bumper always, and parking? forgetabout it). But DC is a must see. I second the Smithsonian...realize that it is a set of museums, not just one. I remember reading many years ago that if you spent 30 seconds looking at every display in the Smithsonian, it would take you 33 YEARS to see them all-and that was before the new museums.......so pick out which ones you want to see. For me, the Museum of Natural History, and the Air and Space Museum are must do's. Also...visit the National Archives. It's not very big or even that popular, but it houses the Declaration of Independence and the Constitution...the REAL ones. There is something very profound to look at the documents that created US. Also, you must see the Vietnam Veterans's Memorial (just ask for "The Wall") and the Lincoln Memorial. Both are very moving.
The National Mall is not a shopping mall. It is the greenswards that starts at the White House and goes a very long way...the Washington Monument, the aforementioned Wall, most of the memorials are there.
The Museums are free, I can't remember about the Zoo. The National Zoo is in Rock Creek Park, which is, as someone else mentioned, not downtown, but it IS in DC. It is definitely worth the visit, though.
Finally, don't be too confused about the fact that DC straddles two states, Maryland and Virginia. the Potomac separates the north side (MD) from the south side (VA). (and the river is pronounced Poh toh mic)
And ...don't be surprised at how small the White House is. You see it on TV and think, wow, it's a big mansion but it is not at all.
