The Washington Post’s Going Out Guide, is a fairly comprehensive listing of what’s going on in the city on certain dates. There is also the official DC tourism site that has information about what’s happening around town. If you’re driving you’ll want to keep an ear toWTOP (103.5fm) for traffic updates. There always seems to be some random parade, protest, game, accident, or construction somewhere to screw up traffic.
My best advice would be to meander a little bit. Don’t always take the most direct or quickest route to your destination. Take the metro bus sometimes (and buy one of the new Van Gogh Smarttrip cards to make it easier to get around) and look out the window. It is interesting to see the way people actually live, in a different city. For instance here, you can’t really say a whole section of the city is bad - you have to go block by block. It’s amazing how different neighboring blocks are.
For specifics, my kids love the Spy Museum, the Newseum, and the Sculpture Garden. The coolest thing about the Newseum (for me) are the daily newspapers from each state displayed in front of the building. If your kids are into irreverent things, in the Columbia Heights area (the Columbia Heights metro stop), there is a neat little shop called The Museum of Unnatural History. It’s actually a gift shop that supports a tutoring program for DC kids, but they have neat things that would make for fun souvenirs. (For christmas stocking stuffers, I bought my kids Mega Sand and aTail growing Supplement.) If you go, next to the metro stop is a Five Guys which has great burgers. Also, if anyone in your family is into anime, there is a neat little place called Coffy Shop [actual spelling] that is filled with drawings of cartoon and anime characters, and they have a shop in the back. It’s about 2 blocks north of the subway stop, on the southbound side of the street.
Also, you might want to spring for a pedicab tour of parts of the city not readily accessible by the subway, like Georgetown. They are a lot of fun, and you get a guided tour of parts of the city that are often overlooked.
If you can’t tell, I disagree with the posters telling you to avoid whole swaths of the city. In my experience, as long as you’re not being supremely stupid (walking around counting gold coins or being frivolous with your electronic devices), nobody really gives a shit about you. It isn’t as if DC’ians are out to get all white person or are mindless zombies with a taste for tourists. Even here, most people are pretty nice. The SITUATIONS you see sometimes might make you a little uncomfortable, but that doesn’t mean those people are going to hurt you. If you take Pennsylvania Ave SE and drive across the Anacostia River, you’ll see that it isn’t a vast wasteland of drugs and poverty and crime. Just like the rest of the city, the neighborhoods vary block-by-block and so do the people.