I have ridden the Natchez Trace in the Nashville area several times and down around Tupelo once. I honestly haven't done much stopping to see the sights but there is a quaint little town near Nashville called Leipers Fork. It's only a couple miles off the Trace so you are not going much out of the way. It has some cute little shops and a popular grocery/restaurant called Pucketts.
Puckett's of Leiper's Fork
Also, about a 1/4 north of Leipers Fork is the General Lee and a police car from The Dukes of Hazzard. Kind of cheesy but something to see.
I would recommend going to the National Park Service website, if you have not already done so. It breaks the Trace up into four sections and lists a handful of places to see and for each section and their mile marker number. Under the 'Plan your Visit' tab, click on 'Places to Go'