First off, let me just say that I'm awesomely impressed with all the folks who want to come to DC for the inauguration. As a long time resident of DC, let me give you some pointers on what to do or not to do while you're here.
Places to stay:
NONE. All the hotels for miles around are booked. If you're lucky, you might find something in Baltimore or Richmond, then you can Amtrak up or down. Yes, I said Richmond.
Transportation:
Generally, GET A MAP, be it a personal GPS or good old paper type. DC is difficult to navigate at the best of times, much less when it's chock full of tourists.
Cars:
LEAVE YOUR CAR AT HOME. Seriously. DC has bad traffic on a GOOD day. And with the inevitable road closings and security, etc, you'll be lucky just to get as close as Dupont Circle or Alexandria with your car, much less find parking. Plus, DC drivers are insane at the best of times, and DC is a total pain in the ass to navigate, especially by car. Hardly a week goes by that I don't help some poor lost soul wandering around with a map looking lost. So save your sanity and dump the auto.
Metro:
A good option, but expect long, long lines when it's time to leave. It takes about 1/2 an hour to clear out the local metro station after a Caps game. Multiply that by the number of people expected, and you might be able to get into a subway car the day after the event. It would probably be better to walk to a subway station in the OPPOSITE direction that you want to go, since all the local stations will be wall to wall people.
AMTRAK/MARC to Union station
Union station is a fair walk from mall, but not undoable if you're able bodied and don't mind the cold. Or you could metro there, but be forewarned that the metro is going to be insanely busy. Be sure to check the schedules online.
VRE to L'Enfant Plaza
L'Enfant plaza is fairly close to the mall. This would be a good option for those coming in from Virginia.
FOOD:
Most of the Smithsonian restaurants are overpriced. Undoubtedly there will be lots of vendors there selling corn dogs on a stick and other carnival fare for a healthy profit. Luckily, not too far up 7th St NW there is a broad variety of restaurants, and Chinatown (such as it is) is at 7th st and H NW. Most of the restaurants close to the Mall are of the upscale type, such as Fogo De Chao.
What to do after the inauguration:
Go see the Smithsonian. You will not regret the time you spend there. I'm not exaggerating when I say it's the greatest series of museums in the world. And please, toss a few bucks into the donation bin at the doors. Admission is free to every Smithsonian building. If you're staying anywhere near Dulles, the Udvar-Hazy Center has a ton of historic aircraft, such as The Enola Gay, the Enterprise, and a Concorde. And yes, it's HUGE.
The Spy Museum is very cool. Lots of neat spy gadget stuff. It's not part of the Smithsonian though, so you have to pay to get in.
The Holocaust Museum is a bit grim, but another excellent museum. A must see for our kosher brothers and sisters, and any 20th century history buff.
Dc is the monument capital of the US. So much so, in fact, you could spend all day walking around looking at monuments and not even catch half of them. Some recommendations: Jefferson Memorial, Lincoln Memorial, FDR Memorial, and the Vietnam Memorial. All are moving in their own way.
If anybody has any specific questions about DC, lodging, travel, etc, I'll answer as best I can.