I live in Montgomery County Maryland, so some of this would be pretty far from Virginia, but I do a lot of traveling around the area with my 5 and 3 year olds, and there's some great stuff for little ones around here!
We love the National Zoo, and other than the parking fee it's free! Very close to the Metro too, if you don't want to deal with parking at all. There's a nice zoo in Reston, VA too, but there is a fee to get into that one.
The Udvar-Hazy Air and Space museum out by Dulles airport is great for little ones. Plenty of space to run around and lots to look at. My kids don't care about all the details, they just like all the different colors on the airplanes! Cool observatory too, where you can watch planes land.
Wheaton Regional Park in Maryland has a carousel and train ride. They're rebuilding the playground, but it looks like it'll be fantastic in the fall when it re-opens! Glen Echo park also has a carousel, and Cabin John Park has a train ride. I think there are a couple of parks in VA that have those too. Look into Burke Lake Park maybe. There's also a great playground in McLean called Clemyjontri (I think that's the spelling) that's wheelchair accessible but great for all kids, and there's a carousel there as well.
For summertime, there are several spraygrounds and water parks around. There's one called Atlantis in Bull Run Regional park, one in Germantown MD that's great for little kids, Turtle Park in DC (near American University).
Jammin' Java in Vienna has daily weekday morning concerts for little ones. On weekends they often have special kids' performers too.
That's all I can think of off the top of my head, but hopefully it'll give you a good start!