That depends on your interests. The I-40 route is about 3 hours of boring highway driving with little of interest except the one bright spot of Loretta Lynn's ranch at Hurricane Mills. http://www.lorettalynnranch.net/main/
For a more scenic drive that will cost you about 1.5 hours in drive time, you can drive the Natchez Trace from TN-100 on the west side of Nashville down to US-64. The Trace is a beautiful parkway maintained by the US National Park Service. No commercial traffic is allowed so no big trucks. Just past Savannah, TN off US-64 you'll find Shiloh National Military Park, where the Civil War battle of Shiloh occurred. It was a short battle lasting just a a few days so it's easy to understand how the battle was fought, as opposed to a long siege like Vicksburg, MS where the location of troops changed over several months. You can go back up to US-64 and continue on it to Memphis, or go to Corinth, MS and take US-72 to Memphis. Either is a good choice with almost no time difference to either route. https://goo.gl/maps/UmVJ23tzEBS2
If you're HUGE Elvis fan, you could continue from Corinth on US-45 down to his birthplace at Tupelo, MS then use Interstate 22 up to Memphis. https://goo.gl/maps/e6KPRTAaAQS2 This will take another hour of driving for a total of about 5.5 hours, plus time for stops.Edited: 07 June 2018, 04:43
My wife and I just went to Memphis and back to Nashville over the weekend. At the current time, I-40 is a mess with a lot of construction, and there were four major wrecks. Consult the Tennessee Department of Transportation Smartway traffic site at: https://smartway.tn.gov/traffic/…
I agree with the previous poster that recommended taking an alternate route to and from Memphis. We were in a hurry to get there when we left Nashville and ended up taking 4 1/2 hours instead of 3 because of the four delays, including two of 30+ minutes.
When we returned to Nashville, we took US 64 to TN-100, and we actually made it back to Harding Road in 3 1/2 hours. There were virtually no other drivers on our route, except in Henderson, Centerville, and Fairview. It was so much more beautiful than boring I-40.
Since we have been back in Nashville, the last three days have been just plain awful for interstate traffic in the city. I-40 West was totally stopped between White Bridge Rd. and Charlotte Pike Tuesday night. I-65 was totally shut down twice in two days, and Briley Parkway was at a standstill.
I drove that route I40 for 3.5 years and then some. There are not really any must do or must see's that are quick stops between Nashville and Memphis. I would recommend concentrating on Memphis and Nashville with such a short time in each.
We did a wonderful Nashville to Memphis trip last summer leaving from Nashville via Natchez Trace Parkway to Muscle Shoals, Alabama (tour historic recording studios, beautiful river park, tour Frank Lloyd Wright home) to Tupelo, Mississippi (quick stop at Elvis birthplace, get out take a picture, free, 10 minutes tops) to Oxford, Mississippi (would go back just to eat - best food, quaint town square, William Faulkner Rowan Oak home) to Memphis and back to Nashville via I40. We did this as a round trip from Nashville from Friday afternoon to Sunday. Beautiful drive (except like everyone says the boring part on I40 back to Nashville).
You don't say how old your kids are but Memphis has a lovely zoo. Another thing your kids might like in Memphis is the Bass Pro Shop. I know there are Bass Pro Shops everywhere but this one is in one of the largest pyramids in the world. That makes it a little more interesting than the others.
I grew up about halfway between Nashville and Memphis and have to agree with the others. There isn't a lot to see in between the two cities. It's a very boring drive but there are some fun things to do in both cities.