Congrats on the move! I love Williamsburg, in fact we'll be there for almost three weeks over Christmas. Planning to do any of the Colonial Christmas stuff?
For DC, pack your best walking shoes! It's easily twice as much walking as a Disney vacation, and mostly on hard marble floors. I'd say for a quickie first trip, concentrate on the Smithsonian. You could easily get lost in a single museum for weeks, but it's possible to see the majority of the highlights in one museum in half a day.
What are your interests? Government in action? History? Nightlife? The arts? I've spent tons of time in DC, but it's hard to give specific recommendations without knowing anything about you
Not the most economical choice, but I would recommend staying at the Gaylord on the National Harbor (actually in MD) and getting an ICE! package and immersing yourself in the Christmas festivities that hotel has to offer!
We only live a short drive away, and are considering staying there this weekend just to get away. Our other option we're considering is actually going to Williamsburg for the weekend.
Everything takes place in the historic district (run by the Colonial Williamsburg foundation). There's an admission fee, but annual passes are dirt cheap. Of course, a lot of the special programming costs extra 
Christmas starts with the Grand Illumination December 5. No admission fee, it takes place on the Palace Grounds, but bring a blanket or lawn chairs and get there early in the day to stake out a spot. Our favorite special event is the Yuletide Supper at the Williamsburg Lodge on Christmas Eve. It's probably sold out by now, but there's a 3 day cancellation policy, so you may be able to get something last minute. Second favorite is Christmastide at Home, a collection of scenes of Christmases past. That repeats a lot and is easy to get tickets for. Everything's worthwhile though.
