We've driven through Memphis twice through the years and did Memphis in about 2- 3 hours each time mostly by car and doing drive bys. We aren't big Elvis fans, but thought it would be cool to drive by to say we've seen it so we drove to his house and parked along the street (many, many tourists there- area to quickly park) and walked along the front sidewalk area, took a pic in front of the gates, and looked at the house from behind the wall. Quick stop/ check off for us- felt safe enough but went midday and there were tons of people around. We also went by the Lorraine Motel where MLK, Jr was assassinated. It's a civil rights museum inside now, but the outside of the building and location where he was killed are preserved and there are things to see/ read / listen to outside there that are free. It was a good history lesson for my kids who this past trip were all old enough to know about MLK and what he did/ stood for. We spent about 20 minutes at the outside part there- free surface parking right next to it. Besides that we went to the visitor's center on the Mississippi River, drove down Beale Street and by a few other areas before heading out. We missed Mud Island years ago and had looked forward to checking it out this past summer, but it was undergoing a major remodel/ revitalization then and was closed. Check to see if it's up and running. Probably would be fun with kids if it was.
We've been through Nashville twice in pretty much the same way- a few hours with quick stops by car. Most recently we went by the Grand Ole Opry, Ryman, the capitol complex, drove through downtown, by Vanderbilt's campus, and explored the Parthenon and Centennial(?) Park there. If you like country music, the Hall of Fame is there too. We stayed the night before in Franklin- cute little town/ suburb south of Nashville.