Welcome, Tom. Preemption means that while Arlington may be out of line with much of the rest of the state with it's political philosophy, it has to operate under the same rules as every other jurisdiction when it comes to gun laws. Google "Dillon Rule" for a full explanation of preemption.
You can own and open carry a handgun at age 18. No permit is required for OC. You can't purchase a handgun from a dealer at this age, but may in a private purchase or as a gift. There is no registration of guns or their owners in Virgina.
Concealed carry requires a permit which you can get at age 21. You will require proof of training as part of the application process; there are several instructors on the forum. Virginia is a "shall issue" state, so as long as you have no reason to be denied, you will get your permit.