I think MARC or Amtrak, whichever fits your schedule, is the best bet for getting into the city. Both terminate at Union Station, and it's a quick metro ride to Farragut North on the red line. There is a kiosk at the BWI station for buying / printing tickets for Amtrak / MARC. Consult the schedules on their websites and pick out two or three trains that work for your scheduled arrival. Remember also that Amtrak tickets are easily refunded if you don't first print the ticket (if you do print it, you have to surrender the unused ticket to get a refund). So, no harm in buying a ticket in advance and refunding or changing if you misconnect to the train.

If your only two destinations are White House and DCA, you should only need a metro fare card with about $5 on it. If you might make more use of metro during the day, and are arriving at Union station after 9:30 am, buy an all day pass for $9. Metro has a useful trip planner: http://www.wmata.com/rider_tools/tri...ripplanner.cfm

Trains during the day can have a bit of lag time between them, and are often impacted by maintenance work. On departure, you should take the blue line from Farragut West (not North! like your arrival) or McPherson Square station toward Franconia Springfield. Travel time is about 15 to 20 min once you catch the train, but you might wait as long as 15 min to catch a train if luck is not on your side. I'd leave at 1:50 if going by metro. If you're looking to maximize your time in downtown, you should be ok to hop in a cab by 2:15. ~$20 fare. Obviously things can go really wrong, but it rarely takes me much more than 15 minutes to get to DCA by cab in the middle of the day. If you don't already have a BP, and can't use a mobile BP there are a handful of standalone kiosks that are along the walkway opposite of the ticket counter. There's never a line to use these. If you need to check a bag, then of course you have to go to the ticket counter (there's also a US bag check & ticket counter near the metro entrance, which often has no wait).

A car is generally a bad idea. DC can be tricky to navigate for the novice, inbound traffic from Maryland can be horrible (though generally lighter in July/August), and I'm familiar with a lot of fun stories about GPSs gone wrong in this town. Garage parking in the downtown area will run you from $20 to $30 depending on the garage. There is an AVIS on M Street NW around 17th. You could do a one way rental to there, drop it off upon arrival, walk down to the White House (6ish blocks), and then cab to DCA later for the same $20 you'd pay to park the car.