Begin by taking the Metro blue line to Arlington Cemetry. Walk around the cemetry, a map is available from the visitor's center. The climb up to Arlington House, gives a good view of the city and is near to John F. Kennedy and his brother Robert's graves. You can watch the guard at the Tomb of the Unknown Soldier too. When you've finished, walk back towards the Metro but walk over the Memorial Bridge into D.C. Infront of you is the Lincoln Memorial.
From Lincoln and facing the Washington Monument you should take the left paths to lead you to the Vietnam Memorial, a long black wall inlaid with the names of those that lost their lives.
From the Vietnam Memorial, make your way around to the Korean Memorial, which includes an engraved wall and full size iron cast soldiers - an eerie experience.
From here, cut right across the mall lawns until you reach the Tidal Basin, follow the path around the Basin and you'll come to the FDR Memorial. This is a modern memorial with waterfalls and pools, it is laid out like rooms of his house and has many interesting features. There is a visitor's center and park rangers to guide you through it.
Continue along the path around the Basin and you will come to the Jefferson Memorial, which offers a view across the Tidal Basin to the South Lawn of the White House.
From the Jefferson you can take any number of paths or streets, I usually use 14th street and arrive back in town by way of the Bureau of Engraving and Printing which offers a good cafe if you're flagging at this point. Then it's back up to the Washington Memorial.
At this point you can finish your walk or you can continue along the Mall to the Smithsonian Castle, the visitor's center for all of the Smithsonian Museums. Further along the Mall there are sculpture gardens to distract you or simply continue on past all the museums until you reach the Capitol Building. A climb up the steps to the top will give you a good view of your walk.
To finish the nearest Metro to the Capitol is the Capitol South Station, signposted from the Capitol.