I would like to go from Georgetown, MD (near Galena, not near Washington) to Cape Charles, VA next week, without a car. What's the best way to get there? What are the possibilities?
Well since you've not been too specific aside from 'without a car', I'll take this opportunity to go to town.
1) It's technically possible to walk between the two places. According to Google Maps and Directions, it'll take you approximately 1 day and 15 hours of solid walking, so I'd suggest trying to fit in rest stops, at least in Baltimore. Also be aware that some of the roads may be cars only and pedestrians forbidden.
2) Flying - it looks like the nearest decent sized airport is Baltimore, which is half way there really. However, I can't find any direct flights, so you may need to go via another center, really making this not a very logical option unless you enjoy queues and the TSA experience.
3) Hitchhike - In most states you can't hitch from the interstates (motorways) themselves, but you can always stand at on-ramps (highway entries) like in Europe; if there is a "No hitchhiking/pedestrians" sign, don't stand or hitch from beyond the sign. In some areas (such as certain towns or municipal areas) hitching is illegal everywhere, however, it is still allowed de facto. The police in a region may interpret laws related to hitchhiking differently, at times forcing a hitchhiker to choose an alternate route by walking or using other means of transportation. In most cases, though, hitchhiking is legal or tolerated as long as you are not on the interstate itself, where it is rightly considered a safety issue. There are also many limited-access highways (i.e. with on-ramps and off-ramps) that are not part of the interstate system; these typically prohibit hitchhiking as well (other than at the on-ramp).
4) Other than that, renting a car is always an option if you just meant you don't personally own one...