Arlington Soccer does the rec league for the county. They run in the fall and spring and they try to place kids on teams roughly based on geography/school. Spring season is already underway, so look for sign ups for fall around June. The time commitment is one practice on a weeknight (1hr) and one game (1ish hour) on the weekend. All games and practices are in the within the county. Ignore any travel options for now, they are much more time and money, see if the kids like soccer first.
https://www.arlingtonsoccer.com/
In the winter, Arlington County runs a house basketball league for 3rd-12th grade and a weekly skills class for 1-2 grade. Sign up is in late August. Again, teams are based loosely on geography. The time commitment is generally one practice and one game per week.
https://www.arlingtonva.us/Government/Departments/Parks-Recreation/Programs/Sports/Basketball
There are also options for softball, gymnastics, and any other sport you can think of (archery even!), but these are the leagues I'm familiar with. They are both fine for beginners. If you happen to know a friend is already on a team in one of these leagues you can ask the coach's name to try to get placed together.