Getting from Union Station to National Mall?

Exactly.  You'll see the Capitol Visitors Center from Union Station which sits at one extreme of the National Mall, so it's just a few blocks walk to that end and a nice walk though it. If you'd rather Metro, you'll exit the train portion of Union Station to the Metro side on one end. Take the Metro to the Smithsonian Exit which ends near the  Smithsonian Castle Visitor History and which is closer to the  Washington Monument. The metro option will get you closer to the other memorials like the  National World War II Memorial and the  Lincoln National Memorial.

It is a 1.2 mile walk from Union Station to the National Air and Space Museum on the Washington Mall.  Walking there, you will pass by the United States Capitol, so be sure to walk south on either Louisiana Ave or Delaware Avenue and either of those streets will dead-end on Constitution Avenue - where you will be able to see the National United States Capitol on your left, and the National Mall immediately in front of the Capitol Building.

The Metro in the District is clean, fast and efficient and as has already been pointed out, you could easily ride it to the Smithsonian Exit on the Mall, but then you would miss the nice walk !

Have fun !

You can easily walk or take the DC metro to the smithsonian stop via the red line connecting at metro center to the smithsonian station.

You can use the DC Metro line from Union Station.  

The easiest way is to take the Red Line (at Union Station) and switch to the Blue Line at Metro Center just one stop over. Once you reach Metro Center, take the Blue Line towards Franconia. You will exit at the Smithsonian stop. There are two exits at the Smithsonian stop-- one that is directly on the National Mall  (which features the Lincoln National Memorial, Washington Monument, Museum of Natural History, and the National Museum of American History). Additionally, this route can take you towards the Fdr Memorial, Washington and MLK Memorial. The other Metro exit leads to the United States Holocaust Memorial Museum. USHMM is a bit of a walk, but arguably something every adult should see.

The DC metro is very easy and the maps you'll find there are some of the easiest of any city. Take the metro directly from  Union Station to Judiciary Square Metro Station and it's a quick five minute walk to the mall from there. 

It is actually walking distance so you could just hike, there are buses too right out

Get on the metro at Union Station and ride it to a stop near the National Mall. http://www.wmata.com/rail/maps/map.cfm?

DC Metro

Union Station is basically at the "top" of the National Mall. You could easily walk, depending how far down the mall you want to go. There is also Capital Bikeshare. But no good direct Metro options. A taxi to somewhere mid-mall would cost less than $10.