There are several ways to get from the white house to Arlington cemetery.: plane, train, bus (Greyhound) and car. The easiest and most convenient (takes only 15 mins) is on the Washington Area Metro, which you can get from Union Station in Washington DC. Otherwise, you can catch a flight from Washington Dulles Airport to Ronald Reagan Airport.
You can walk to the Arlington Cemetery from the Lincoln Memorial...it only takes 30 minutes walk. Once there I suggest joining the Tourmobile bus tour because the cemetery is huge plus the bus will take you to all the major spots where you can get off.
Parking is pretty tight anywhere in DC, so it's always smart to to take the Metro or use other types of public transportation. When we went to DC, we flew into Reagan and parked at one of the hotels near the Metro in Maryland. We never did park in DC and we were able to see lots of things. When I got tired of walking, I just hopped on a bus and paid a buck or two to get to Arlington.