Hi, I am originally from Memphis but live in Central Florida now - we have done this drive many times a couple of different ways...
I am not the best at giving directions, but I'll give it a try:
You could take US-78 to US-280 to I-75 to SR-528 (what we like to call the "MS-AL-GA route") or you could take I-40 to I-75 to SR-528 (what we call the "Nashville-Chattanooga-Atlanta route"). Also note that once you are in FL (on I-75), you could take the interstate (I-10 I think?) that goes over to I-95 and then drive down that to SR-528 or FL-528 (whatever they like to call it).
Either way it is a LONG HAUL! The quickest we have done is probably around 13 hours but we don't have kids...
Whatever you decide, travel safe!