Alternate route avoiding I-95

Ok - here is a revised trip (not dealing with a dog, but just listing routes) From Reading (Pennsylvania) take I-76 over to Harrisburg and then take I-81 down to Blacksburg (Virginia) for the first night. That's about 6 hours driving on the interstate. You could also take the Blue Ridge Parkway for part of that trip.  The Blue RIdge is a nice drive but there aren't any services on it and it takes time to get to it and get off of it.
From there take I-81 S, I-77 S, and I-20W down to Augusta or Macon It will be about 5 hours driving to Augusta and probably 6 to Macon 
From Macon take I-75 all the way down to Ocala (this is a fairly peaceful drive for an interstate) and then cut over to Daytona Beach  It will be 7 hours from Augusta and about 6 from Macon.

Speaking only for South Carolina and points south, the drive from Little River to Beaufort is quite nice. The Grand Strand is somewhat hectic with tourists but Georgetown (South Carolina) is a nice spot to stop with a few inns in town. Or, depending on time, Charleston is only another hour down 17.  Past there you have a great drive through the Lowcountry into Beaufort and do not have to get on I-95 at all. You can stay on secondary roads into Savannah and reconnect with 17 all the way to Jacksonville. There you could get on A1A. This is not the fastest route but it is scenic and filled with Americana. 

Easy - I really don't like I-95 so I only use it if I am in a real hurry. 
From Philadelphia you could go down to Cape May and take the ferry to Lewes.
Or you could just take the NJ Turnpike or I295 and get off in Delaware.  Then take Delaware route 1 and/or US 13 down the lenght of Delaware to Maryland and take the Chesapeake Bay Bridge Tunnel down to Norfolk/Virginia Beach/Chesapeake.  Stop the first night in Elizabeth City (about 330 miles about 6 hours of driving - some tolls).
Second day follow US 17 down the coast to Georgetown (South Carolina) (about the same distance). 
Third day follow US 17 down past Charleston. US 17 cuts over to I-95 and that will take you to Daytona Beach.

I think the key is taking the route along the coast as RosalieAnn mentioned, especially if you have 3 days to drive.
If you start in Reading (Pennsylvania) instead of Philadelphia, it's possible you may want the first part of your trip to go faster since your local area is probably less interesting to explore. So you could do US-222 to US-30 then down I-83 to Baltimore.
If you get all the way to Baltimore on major highways, then you could try to get to US-50 to cross over toward the coast, then follow Rosalie's directions from there.
Really though, from Reading it almost makes more sense to just go toward Philadelphia or Wilmington (Delaware), then just stay closer to the coast all the way down.
It just depends on how you want to time each part of your trip.

If you are starting from Reading and are going to use I-85, what other information do you need?