Visit the US Capitol - it's a very short walk up Capitol Hill to the Visitor's Center. Timed tours are available by reservation, which can be made on the internet. Walk up tours are also available, but in the busy season (Spring is the busiest, Winter time is the most slow) the wait can be LONG. Tours are free.
Go for a walk - Walk straight out the front of the station and you'll see the Capitol in front of you - go just to the left of it and you can walk down First Street NE/SE between the Capitol grounds and the Supreme Court (NE) and the Library of Congress (SE). Once you get past the Capitol, hang a right and walk down the mall as far as you would like. The Air&Space and American Indian museums will be right there if you want to wander in, or you can just loop back towards the station. The National Archives is also right there and worth peeking into quickly.
Here's a map, this route is 3 miles, but you can shorten it considerably:
There are a number of hop-on, hop-off type vehicle tours that use Union Station if you'd like to venture a little farther afield. I'll defer to others that have used them.
If you're interested in seeing train type stuff, all the lines of the MARC (Penn, Camden and Brunswick), both lines of the VRE (Fredericksburg and Manassas) and the Washington Metro (Red Line) all stop at Union Station. The Penn Line runs on the NEC up to Perryville, MD and allows stops at local stops that Amtrak doesn't serve. It has the most frequent service, so if you want to spend a little time on the NEC for cheap, this is the way to go. The Camden Line also serves Baltimore, but runs along the CSX freight route and terminates at Camden Station (home of the Orioles). It's possible to do some same day turns on the Camden Line if you time things correctly. It's also possible to use the Light Rail in Baltimore to transfer between Camden and Penn stations to do a loop trip if you desire. Finally, the Brunswick line follows the route of the Capitol Limited as far as Martinsburg, but also has a branch that diverges at Point of Rocks that serves Frederick, MD. On this line trains run inbound in the morning and outbound in the afternoon, so you can't do a same day turn. You can however take the Brunswick line outbound as far as Rockville and transfer to the Metro Red Line and then use that to return to Union Station in the afternoon.
I think that VRE also has some reverse peak service, but I've never ridden it.
There's plenty more to add, I'll try to get back here periodically and put in more information of interest.
Edited by Ryan, 21 April 2011 - 08:02 AM.