In the example we've already found the Holocaust Museum and decided we're going to the Smithsonian Station. Let's say we're starting out at the Rockville Station. By looking at the map you see that Rockville is on the Red Line (upper left), and Smithsonian is on the Orange and Blue lines (around the middle of the map).
Click the above map for a larger version. See below for a zoomed version showing our stations circled in red.
Rockville and Smithsonian stations
Uh oh. Did you notice our first complication? Since Rockville is on the Red line, and Smithsonian is served by Blue and Orange lines, we're going to have to change lines. Don't panic, because metro has conveniently highlighted stations where you can transfer. See the big double circle two stations above Smithsonian? That's not a carousel, it's a a transfer station: Metro Center.
We're armed with our third clue, our transfer station. Now, how to get there?
Trains running throughout the system are named for their ending station. For the most part, that's the last stop on the line. You'll see Orange line trains either going to Vienna (out on the western end) or New Carrollton (eastern end). Red line trains will be going to either Shady Grove (northwestern end) or Glenmont (northeastern end). However, red line changes up a bit at rush hour. On the red line about half of the western side end at Grosvenor, and half of the other direction end at Silver Spring. But you get the gist. To head into the city from Rockville, you'll take a train that says either Silver Spring or Glenmont. They'll be on the same track heading in the same direction, making all the same stops (until one ends early up on the other end).
When you get to your transfer station, you'll want to know what train to find. From Metro Center you can take orange towards New Carrollton OR blue towards Largo Town Center. Both lines will get you the two stops from Metro Center to Smithsonian.
Congratulations. You now know all your stations and the correct trains to look for! Next up, financing your journey.