ALERT FOR MISSIONTRAVEL -- I fear that McCarthys5's DC memories are fading. He/she has confused National Cathedral (properly the Episcopal Cathedral of Washington), at Wisconsin Av. & Garfield St. NW, with the National Shrine of the Immaculate Conception, a Catholic basilica on or adjacent to the main campus of Catholic University of America,in Northeast DC.
To Georgetown from the Cathedral neighborhood, there is no reasonable Metro route. Metro is a radial system connecting DC and its suburbs that does not provide much in the way of intra-DC transportation. Take almost any bus straight down Wisconsin Avenue, and get off anywhere between R St. and M St. NW, and there you are in Georgetown -- about 10-15 minutes. Taxicabs are pretty cheap, and the best bet late at night (buses run infrequently at night, and the last bus may be earlier than you're ready to turn in).
Choosing between Georgetown and Old Town Alexandria is hard (as a loyal Alexandrian, I must mention that Alexandria is two years older than Georgetown, founded in 1749 vs. 1751). It depends what you want to do, and maybe how old you are. Georgetown is lively much later into the night, with lots of bars including some with trendy music. It also has lots (swarms) of high school kids, lots of T-shirt shops, etc. All the Georgetown restaurants mentioned are excellent -- but be advised that you'll need reservations well in advance at most of them, especially on weekends.
Alexandria also has lots of good restaurants, but is much more sedate in the evening. It's also a nice place for sightseeing, while G-town really has nothing in that line. The two best in town are Eve (as good as any DC restaurant, pricey, you do need to book in advance) and Elysium at the Morrison House Hotel (same). Other good ones are La Bergerie (classic French with some recent updating of the menu), Le Gaulois (provincial French), Villa d'Este (North Italian, out of the main tourist area but still in Old Town and my personal favorite), Le Refuge (French and tiny), the Majestic (New American), Bistrot Lafayette (French, of course), 219 (New Orleans, may have good jazz). Landini Bros. (North Italian), The Wharf (seafood). A good down-and-dirty chili parlor is the Hard Times Cafe.
To get to Alexandria (King St. Metro), in addition to the Yellow Line Metro (coming down 7th St. NW in DC) mentioned by another poster, you also can take the Blue Line Metro, coming across NW DC from Union Station through Metro Center (the main hub at 12th and G NW) past Farragut Sq. (Connecticut Av. NW) and Foggy Bottom (Geo. Washington Univ. area) to Rosslyn, Virginia and then down past National Airport to Alexandria. It's quite a hike from the King St. Metro to the heart of Old Town, maybe 17 blocks, and the heat/humidity these days is ghastly. From the Metro station, take the DASH bus, and depending when you get there you could take the free DASH-About shuttle bus into Old Town. Lots of taxicabs at the King St. Metro, cost to the heart of Old Town about $5.
If you'd like to stay in Old Town, the Morrison House Hotel is world-class (and priced accordingly). Alternatives are the Holiday Inn Select in the heart of Old Town (large and bland high-rise but comfortable enough) and, a few blocks away from what action there is, the Ramada Inn on N. Fairfax St.
Post again if you have other questions or want more detail, and have a great visit.