We visited DC last year for a couple of days and stayed in Arlington, VA at a Sheraton. We got a great rate through hotels.com.

We were nervous about using the Metro for the first time but it is really quite easy. The station stops are well marked. We went to Smithsonian also. I don't remember the stops but when you get to the metro you should find someone there to assist you. You purchase your tickets through a machine. There was an attendant there who came over to the machines and helped us purchase our tickets and which stop to get off at. It's really quite easy. My husband and I are seniors also so if we could do it anybody can. LOL Have a great day. Smithsonian is awesome. We did the National History Museum also and loved that.

Amtrak has a stop at BWI airport which is about 18 miles from Annapolis. About 31 Amtrak trains go from BWI to WAS everyday.
The Acela trains take 7 less minutes from BWI to WAS which does not justify the extra fare IMHO.
Union Station is on top of a major METRO station. Are you staying in the city or just going to DC for a day trip.
You probably need to find a bus from Annapolis to BWI.

Keep in mind "The Smithsonian" is actually a group of museums that include Air & Space (both on the Mall and Udvar-Hazy near Dulles airport), Natural History, American History, American Art, and American Indian to name a few. Go to www.si.edu/museums to see which Metro stop is closest to the museum(s) you would most like to visit. The websites for the Newseum, the Spy Museum, and the Museum of Crime and Punishment will give you maps with the closest Metro stations along with operating hours.

Largo Town Center is the last stop on the Blue Line in Maryland. You can take the train all the way to the Smithsonian station which will get you 5-7 minutes walking distance to:

Smithsonian - at the west end of the mall, closest to Washington Monument, American and Natural History museums, Holocaust Museum, Printing and Engraving

Other station stops in the vicinity will get you slightly closer to these places:

L'Enfant Plaza - Air & Space Museum

Archives/Navy Memorial/Penn Qtr = very close to Newseum, Archives & Navy Memorial. Probably the closet stop to National Gallery of Art (and you can go from east to west wing indoors)

Gallery Place - spitting distance from National Portrait Gallery/American Art, Spy Museum, Building Museum

All the places you mentioned are very walkable although I get it that your parents are elderly and the heat has been fairly brutal lately. Take a look at the metro map. It's not very complicated but check google maps too b/c the walking distance can be shorter. Also check the Metro website for planned maintenance and outages for escalators and elevators.

The Metro system web site has a trip planner that should help you answer your questions. See:

http://www.wmata.com/rider_tools/tripplanner/tripplanner_form_solo.cfm

The Archives/Navy Memorial and the Gallery Place metro stops are on the Green/Yellow line. You would have to get off the Blue line train at L'Enfant Plaza and hop on a green or yellow train. to get to them.

All trains are designated by a color, and the name of their final destination. Thus, the train you get on at Largo Center to get downtown will be labeled as the "Franconia-Springfield" train. At L'Enfant Plaza, catch a green or yellow train heading toward either Fort Totten or Greenbelt. The Archives stop is the first stop you will come to; Gallery Place is the second.

Another great website is:

http://www.stationmasters.com/

Click on the colorful icon next to "Metrorail System Map". This will bring up a nice , general map of the system. Then, clicking on the individual station name on this map will bring up a very detailed map of that particular station including the specific locations of regular entrances (black boxes with white arrows), elevator entrances (bold, black "U" shaped symbols), and the names of buildings and attractions around that metro station.

Don't hesitate to hail a taxi to shuttle you six or eight blocks if the elders are wilting in the heat. Fares would probably be in the $10 range for a party of 4 to travel a mile in standard traffic (Metro fare x4 would be $7ish).