Between Union Station, Washington and Baltimore you can take the MARC commuter train, either the Camden line
http://www.mtamaryland.com/services/marc/schedulesSystemMaps/camden.cfm
or the Penn Line
http://www.mtamaryland.com/services/marc/schedulesSystemMaps/penn.cfm
The Camden line drops you near the baseball stadium, not too far from the inner harbor (a popular tourist area). The Penn line goes to the Baltimore train station. Both of these are weekday (M-F) only. On weekends you'd have to use Amtrak.
The Dulles website says you can take the "Washington Flyer Coach" to get you onto the Washington Metro system at West Falls Church, and then take Metro to Union Station. You're going to have to change trains if you do that as West Falls Church is the Orange Line and Union Station is on the Red Line.
http://www.metwashairports.com/dulles/parking_transportation_3/dulles_travel_options
They have a number of other suggestions for getting into DC also.
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