Washington has a famously efficient Metro system. There are plenty of stops, it’s not expensive, and it’s usually faster than taking a cab. Getting a SmarTrip card is a good idea if you're planning to take the Metro a lot. It's plastic and fits in your wallet just like a credit card. In addition, it's very easy to use.
Depending on your skill, having a bicycle in Washington can be a very handy alternative to public transport. Cycling is common, but beware of bicycle thieves, also very common. A good u-lock and a proper bike rack ought to keep your bike safe. George Washington has bike storage in their basement, and bicycles are also allowed in rooms (assuming your roommates don’t mind). If you don't want to buy a bike, consider using Capital Bikeshare - you can borrow a bike, ride it to where you want to go, and park it at any of their 300 stations around the city.