@Cheyann I will say, I'm not too sure what there is to do in Lake City, but in Tallahassee there's the Tallahassee Museum (kinda like a small zoo) and a couple parks and play rooms. Gainesville has a butterfly museum that my little sister LOVES. I'm sure there is more there, but I don't go to Gainesville all that often.
Unless you want to go to a State Park, there's not really anything in Live Oak. I'd suggest traveling to Lake City (30-ish minutes), Tallahassee (90-ish minutes) or Gainesville (75-ish minutes).