Washington D.C. is a large enough city that you should be able to find an apartment at any time of year. Be aware that apartments closer to the universities will be in higher demand, and you should start looking in the summertime when other students are vacating those apartments.
If you're looking for an apartment in Washington D.C. you should focus on housing that suits your needs. Students should shop for student apartments near their school, for example. Other apartments tend to be reserved for people making certain salaries, or are in parts of town that are either undesirable or inconvenient.
Driving in Washington is a bit of a nightmare, so most people leave the car at home. Instead, get a vehicle with two wheels and join the thousands of cyclists that take advantage of the city's excellent network of bike lanes. For trips further afield, the Metro can get you to the general area where you need to go, and then most people opt to just go the rest of the way on foot.
The Mall is a must-see for every new person in the city. This part of D.C. includes some of our country's most important monuments, including the Lincoln Memorial, the Vietnam War Memorial, and of course, the White House. A rich museum district lies in the eastern section of The Mall and just to the north. Six professional sports teams keep the sports fans happy. D.C. is also home to one of the best jazz scenes in the country. Head down to the Shaw neighborhood to see fantastic concerts being held in venues that have been played by every great the genre has known.