I'd check the U of M shuttle schedules that will be operating between campus and the College Park or Greenbelt Metro stations, as that will largely guide the effectiveness of public transportation over the course of your visit.
From a public transportation standpoint, the airport option you definitely *don't* want is Dulles (IAD).
From Reagan National (DCA), you could take Metrorail to the College Park/University of Maryland stop close to campus.
From Marshall/BWI Airport, you could take the B30 Metrobus to the Greenbelt Metro Station. From there, it would be a short taxi ride to UMCP, or you could ride one stop on the Metro to the College Park station.
You won't need a car for sightseeing in DC. But having one may not be a bad idea if you're flying into BWI and want to get from/to the airport quickly and comfortably, and you defiinitely want one if you'll be traveling around the suburbs surrounding UMCP. Knowing a little more about the logistics of the week's activities at UMCP and where you'll be staying will get you a better opinion.