Welcome! Lots to do and see in DC. You'll have more than enough (probably too much!) to see in a single weekend. Given those timeframes, I would recommend staying for the downtown highlights like the Smithsonians (e.g., Natural History, Air&Space, etc.), the Memorials (Lincoln, Washington, Jefferson, Vietnam, Korean and WWII). Those alone are two days worth right there! Not to mention checking out the sights in Georgetown, the Iwo Jima memorial across the Key bridge, and some of the other stuff folks mentioned. I will say that heading to Mount Vernon is fun but a day trip in and of itself.
As for where to stay, the hotels closer to metros are generally more expensive, but if you plan to head downtown, the higher prices are worthwhile. Parking is atrocious and you will literally spend hours trying to find a spot (without the guarantee of success!). You may want to consider the hotels in the Rosslyn area of Arlington, Virginia. They are close to the metro and tend to be pretty reliable as well as clustered all around. Still, don't be surprised to hear prices in the $200/night range per room. Unless you head way out, that's a typical price for metro accessible hotels. The Crystal City area of Arlington also has a LOT of hotels (Marriots, Hyatts, Hiltons, etc.)
Check out the subway (known as the "metro" in DC) system maps at www.wmata.com to plan your trip. Have fun and feel free to ask any more questions!