There are four main public shooting ranges in Northern Virginia.
Bull Run Regional Park Shooting Center is primarily for shotguns. They have multiple fields for skeet, trap, wobble trap, 5 stand and sporting clays. They are mostly run by volunteer help, and are always looking for more. Additionally they have tournaments, shooting classes, and archery facilities. Inexperienced shooters are encouraged/required to take one of their basic shooting courses before use.
The NRA HQ Range is by far the nicest public indoor range in the area. It is a 50 yard underground range, allowing most rifle and pistol calibers, and shotguns in certain circumstances. It is bright, well ventilated, and features high-tech, programmable target holders. The only downside is that they lack rental guns for first-time shooters to try out. New users must pass a written shooting safety test, which is open-book and not too complicated. They regularly host competition events of all sorts.
Blue Ridge Arsenal is a gun store in Chantilly which also has an indoor shooting range for both Handguns and Rifles. It is not nearly as nice as the NRA range mentioned above, but still suits its purpose. The staff is occasionally friendly, and the selection of rental guns is extensive, including fully-automatic rifles and sub-machine guns.
Sharp Shooters Small Arms Range in Springfield is small, but the staff is very friendly. Once again, the range is not nearly up to the technical standard of the NRA range, but the helpfulness of the staff more than makes up for it. They also have a large selection of guns to rent, which is helpful when deciding between two potential purchases. It should also be noted that only pistols and pistol-caliber rifles may be used at the range. As of June 2012, rifle rounds up to 30-06 are allowed.
If you're willing to go further afield, Clark Brothers in Warrenton and 340 Defense in Berryville are the closest outdoor ranges. At Clark Brothers, there are no range fees, but you must purchase your ammo there in order to use it. 340 Defense charges $20 per hour but you can bring your own ammo. They also offer full auto, rifle and sniper ranges.