There are three main ways a newcomer finds a metro station: look it up on a map, ask someone (friend, concierge, the homeless man who you actually gave money to), or walk around until you see one of the Metro signs.
geeks enthusiasts call these pylons.
The metro map itself is great for navigation, once you're on the metro. If you want to find out what's actually closest to you I suggest using Google maps (or your preferable mapping source) to find the station closest to your starting address. But just to confuse any unaware tourists, many stations have more than one entrance. To find the closest access to your newly adopted metro station, a more detailed map like StationMasters is the way to go. Go to their Metrorail System map, and click on a station name to see a detailed map of the entrances for that station.
If you've already been directed to the nearest station, good for you. Not everyone has had such friendly encounters.
You can use your new mapping method to figure out where you want to end up. Say you want to go to the Holocaust Museum. Your search comes up with a few nearby stations: Smithsonian, L'Enfant Plaza, and Federal Triangle.
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See the Holocaust Museum on the bottom left? It looks like Smithsonian is the closest station, but it has two exits. Rather than standing on the platform flustered and confused when you get to the station, and nearly being trampled by a hurried commuter, find which exit you want now. On the Stationmasters page, you've navigated to the Smithsonian station and see this map.
It even shows the US Holocaust Memorial Museum (on the left, along Raoul Wallenberg Pl.). It looks like the Independence Ave. Exit is the closest, and the arrow shows that when you come out of the station, you'll be facing away from 12th St, which happens to be the direction you want to walk.
Now you're armed with your starting and ending stations as well as some simple walking directions. Next up, navigating the metro system map.