Silver Spring is right on the metro (subway) which can get you to nearly all of the popular tourist attractions. There's also several bus lines going in and out of Silver Spring. You can get a card that is like a credit card that you add money to that can be used for the subway or bus. It makes traveling v. convenient.
In Washington, I'd visit all of the monuments, tour the white house if you can get tickets, the American History, Natural History, American Art, and Air & Space museums (all Smithsonian museums and free), the National Gallery (also free), in addition to the Spy Museum and the Holocaust Museum. In terms of shopping, restaurants, I enjoy going to Adams Morgan, Georgetown, and Clarendon. Go to Ticketplace and get a half-price, day of show ticket for a concert or play. A nice day trip for shopping is Potomac Mills, which is about an hour south of DC.
In terms of NYC, you can take the Bolt or Megabus for super cheap fares (as little as $1 each way). Go to Times Square, Fifth Avenue for shopping, see a show, etc. For New York, I'd recommend Photo Trek Tours, which is a walking tour of a specific neighborhood. You get a personalized tour, and a CD of photos of your trip.
Weather wise, neither city gets a ton of snow in December, but it will be cold. A warm winter coat, hat, and gloves/mittens are a must. Also bring at least two pairs of comfortable walking shoes. Both DC and NYC are places where you do a lot of walking.