I don't know about free things to do, but I know that the oldest European settlement is there, great historical sights all through downtown, most of which are free I believe. Also, my kids LOVED the alligator farm. It was really neat. I think they could have spent all day there. I would look it up online; it's called - big surprise - The St. Augustine Alligator Farm. St. A is a city that thrives on tourism, so there should be a website et al.
We live right across the Potomac from So. MD and we've made this trip SEVERAL times. I know that California, MD is closer to Pax River though, so you'll likely take 13. However, if you go out to 301 and take that down to 95, I think you'll be happier with your trip (so long as you don't get stuck behind an Amish carriage on your way to 301! lol!). For that length car drive you definitely want to take 95 if you can. Don't forget to stop at "South of the Border" on the N&S Carolina borders. It's cheesy but fun for the kids and it's about the halfway point so it makes a good rest stop.
It takes us about 10hrs. to get to Jacksonville (not counting pit stops), which is where my step-son lives and where my hubby was stationed for a while. St. Augustine is another 30 minutes to 1hr. from Jax depending on how you go. I don't remember the actual routes though. It was several years ago. It's also only about 2hrs. from Orlando if I remember correctly, so Disney World isn't out of the question for a day if you're already going to be in the area (should be cheaper if you go on one of the days after the school year starts). Also, check out Jacksonville. I think there is a water-park there or something. Can't remember exactly, but I do know that there is plenty to do in that area if you look closely enough! lol!
Well, that's all I can offer, HTH.