The best way to come to Nats Park is by subway Green Line, to Navy Yard station - closest entrance to the park is at the back end of the train, coming from Chinatown/Gallery Place and L'Enfant Plaza.  However, the opposite entrance, at M Streets and New Jersey Avenue, is much less crowded after the game, so walk over there rather than trying the closest entrance.  If the game has been crowded - everyone staying to the end and/or a strong attendance to begin with - it may be easier to walk North on New Jersey Avenue to get to the Capitol South Station on the Orange and Blue Lines - change at Metro Center for the Red Line.  If the game has been crowded a lot of people will be walking this way, so it is okay even alone at night.

When you come out of the main subway entrance and turn left, you are looking into Nats Park at Center Field from about a block away.  While you can see the full building and the "main" entrance by driving around from the south side (coming up South Capitol Street) the center field gate is the way to go - Nats Park is the only baseball park that I think of the space being more a void than a building, as you get more a sense of what the building contains than what the building is itself.

Things to see - no matter where your tickets are, do make a point of going to the upper deck to see the views, especially of the Capitol, which seems close enough to touch.  To do this take two escalators at the left field gate - just at the start of the enclosed concourse, coming around from left field.  To get down (at the start of the game) either walk down 10 ramps, or ask where the elevators are.  (During the game the escalators are reversed to go down.)

Seats in the upper deck can be very good, and very inexpensive.  409, which looks down the first base line, also has terrific views of the Capitol.  312 has a good Capitol view, and is below the pressbox, so you also can hear the "voice of God", the official scorer and the PR types, who will give the pitcher's line after he is finished, etc.

A novelty that a lot of people enjoy at the park is the Racing Presidents, and their race always occurs in the middle of the fourth inning. The Presidents will be out in the Center Field plaza and outside the Center Field gates before the games - around 6:15 for a 7 pm game, and so forth.  They are happy to pose for pictures.

In the middle of last season the "circle of honor" was established, putting up names of Hall of Fame members from the Expos, the Grays, and the old Senators (now the Minnesota Twins).  None of the "new Senators" (now the Texas Rangers) made it to the Hall of Fame, but there are three Vishnu Baseball statues in Center Field Plaza, one of which is Frank Howard, who does show up from time to time and is a big favorite around here.  The other two statues are of Walter Johnson and Josh Gibson.

Do not try to stay near the park - the area is being built out, and that slowed considerably with the recession, so nothing much is there as far as services go.  So long as you have access to Metro you can just come on the subway.  A trip for Washington baseball can certainly be combined with lots of other tourism, and there is a lot of info available on that.  If you are taking an adults trip and are not looking to entertain kids, you might look at staying in the Dupont Circle area - lots of restaurants, great places to walk, and both the Kandinsky show on at the Phillips Gallery and the Green Show at the Textile Museum (both local private museums - Phillips is the home of Renoir's Luncheon of the Boating Party) are good, and the National Geographic Museum also always has interesting stuff on. Also try the Renwick Gallery, at 17th and Pennsylvania, part of the Smithsonian with a collection of American crafts.

Final advice - wearing team gear for another team is fine, Nats fans are very friendly (although we are casting jaundiced eyes at busloads of drunk 20-somethings who show up from Philadelphia - there are a lot of Nats fans who refuse to go to the park when the Phillies are in town just because of this).  For most games you can just buy tickets when you show up, although you might look on Stub Hub for bargains for the more expensive seats.  For Mets and Phillies there will be some tickets available at the box office, but you probably should get them in advance if you have a particular seating preference.  (Only Orioles, Yankees, and Red Sox, and of course Stephen Strasburg's start, have been actual sellouts.)  The park is designed so that left field line is North - South, and the right field line is East - West, so seats on the third base side will have the sun behind you in the afternoon and evening.