One of the best things about DC is that it is relatively easy to navigate compared to other world-class urban environments. DC is very walkable but can be hilly; if you like to walk, there are many beautiful urban hikes to take, but in order to see the whole city, you will need to use transportation.
Capital Bikeshare-a very popular bike sharing service that has served DC and Virginia since 2010-offers a variety of short-term passes for tourists (3-day pass will run you about 13USD) and with over 2,500 bikes at convenient locations across the city, it is a very budget- and health-friendly way to see the city.
The metro is clean and generally safe, but can sometimes be a long wait during rush hours and on weekends, when track repair work is being done (check the website, it's very user-friendly, for updates or delays). The prices vary depending on destination and time of day, but most trips will cost between 2 and 4USD. Both airports, Washington Reagan and Dulles International, are metro accessible (Reagan is closer).
The bus system is great, but might be a little daunting for a tourist. The best way to navigate it is to use the wmata.com trip planner tools, where NextBus will tell you, via GPS, exactly when the next bus is coming. Bus fare is 1.75USD and you will receive one free transfer.
Uber has a strong presence in DC, accessible through their app, and while there is a taxi system, they are notoriously expensive and untrustworthy, and traffic can get very bad. But for those times when you just need to take a cab, you can hail one on any major road.