I would suggest taking the metro. Stay at a hotel near Metro Center (the subway stop) if you can, because that will allow you easy access to the Red and Orange lines, which get you just about anywhere.
As far as food goes, it really depends on what you want and your budget. DC is a top tier dining city. Check the Washington Post's reviews, the Spring Dining Guide just came out. Any Tom Sietsema review is a good one. I'd avoid the Washingtonian magazine "Top 50/100" lists since they tend to either provide outdated recommendations or be totally off.
If you want more upscale, I'd really try to hit Central (French-American), Rasika (Indian, according to some, the best in the US), Obelisk (Italian), CityZen (American). If you really want to spend some money and are ok to drive about an hour, the Inn At Little Washington is one of the best destinations in the country. If you want a similar fine dining, tasting menu experience in the city, Citronelle is also great. For reservations to most places, use OpenTable, and book well in advance (like now).
As someone else said, stay away from most sections labeled NE, SE, and most of SW. The only exceptions are in the capitol/monuments area. The way the directions work, the city is centered at the Capitol, so you'll find some museums and such in the immediate streets SE/NE. As a general rule, don't go past 10th St NE/SE, and no further south than Independence Ave (with some exceptions).
The museums will be tough to hit all in that timespan. They are all great, so pick ones that suit your interests. For me, the Natural History, American History and Air and Space are the 3 I'd start with. Others like the art galleries. The National Archives are really underrated, so you might see less of a crowd there. If you're really into Air and Space, rent a car, drive out to Dulles (30-45 mins) and go to the Udvar-Hazy Air and Space Museum. It's a hangar near the airport and some really cool stuff is there, like the Enterprise space shuttle.
On the metro, we stand on the right of the escalator, walk on the left. Don't be that guy that stands on the left. The rest of us do have to get to/from work If it's crowded, the front car is the best place to try and board.
If you actually have time to leave DC, I'd recommend trying to see the Udvar-Hazy Air and Space Museum (mentioned above) and Mount Vernon. Neither one is metro accessible though.
Oh, and if you're flying, try and fly into DCA. It is on a metro line. BWI and IAD are far - $90 and $50 cab rides, respectively, into where you're going.