I'd go I-75 into Florida, I-10 East to I-295 South to bypass Jacksonville, and then I-95 South to St. Augustine. Then into Vero the next morning. This route takes you around Jacksonville, and you miss Orlando entirely...as well as the construction on the Florida Turnpike south of Orlando. Plus you save having to pay tolls to drive slow. Other than Atlanta, the only real city traffic you'll have will be Macon, and that's pretty manageable.
One hour north of Vero is about Titusville, and unless you are lucky enough to coincide with a shuttle launch, there is nothing to see or do in Titusville.
St. Augustine is probably 2 1/2 hours from Vero and easily the nicest stop on the upper East Coast of Florida...but that depends on your interests. If you're a big NASCAR fan, you may want to stop at Daytona and see the track. If you're a Surfer Dood, going a little farther to Cocoa Beach to visit Ron Jon Surf Shop would be required.
If you stop at St. Augustine, go into town and see the oldest city in America (by more than 100 years), the fort, and all the touristy stuff. Stop by the Columbia Restaurant and get two things - the 1905 Salad and a nice cold Ybor City Gold beer. The rest of the food is wonderful, too. Great Spanish cuisine from Tampa! It's worth a stop.