We have driven that exact drive twice, and are doing it again in September. The first time the girls were 3 months, almost 2, and almost 3. We left about 8 at night, drove all night, and part of the next day. We stopped around 3 the next afternoon about 3 hours from disney. (we were right by jacksonville when we stopped). The next year (kids were 15 months, almost 3, and almost 4), we left around 6 in the morning, and drove all day, stopped around 8 at night and were about 5 hours from disney. We stopped somewhere in Georgia.
This year I am not sure when we are leaving yet, I have not decided which way I like the best. We try to push that first day, and get as much driving done as possible, I think we are awfully close to just being able to go straight through. The second day, we get up late, take our time and finish the drive, and are usually at check in at three.
We did not plan on where to stop, we figured with three young girls, it would be easiest to just go, and stop somewhere when it was time. We just make sure to stop somewhere with lots of choices to pick from. We start looking for a place about an hour before we want to stop, that way we can find some place nice.
We love the drive, it for us is as much fun as the time in disney!