My wife and I did the Trace last year in early October and found it surprisingly pleasant. I would suggest you get Glen Wanner's book,
Bicycling the Natchez Trace. Be sure to read the alternative routes in the back of the book. The Trace is nice riding but it can be a bit boring. Going off Trace is nice once in a while.
Some must see places: "The Wall".
Tom Hendrix has built a tribute to his great great grandmother that is astounding. The guy will talk your ear off but it's worth the time.
Stop by
Fall Hollow Campground. Bill Roper prepared us an impromptu lunch and wonderful conversation to go with it.
Plan a stop at
Tishomingo State Park. Lovely area, cool suspension bridge and wonderful rustic cabins.
We diverted over to Vicksburg and had a great time riding around the battlefield. If you go off the beaten path, you can find all kinds of old trench works and forgotten monuments back in the woods.