Is it best to fly drive or take a train from Dallas to Memphis or New Orleans?
I would drive due to you can leave when you want and you can control where you go no flight delays or anything like that.
Unfortunately, there is no Amtrak train service to Nashville. The old Union Station is now a Wyndham hotel and that is available for viewing. The closest Amtrak station is in Newburn, TN. From there you can take the City Of New Orleans north to Carbondale and Chicago or south to Memphis or New Orleans.
I rode this train about 27 years ago. I believe it was about 11 1/2 hrs. long ride. If using Amtrak, there is a bus/train combo, though the bus portion is negligible as it is extremely short, about 15 mins. The train ride is listed as 9 hours 7 minutes on the Amtrak schedule, which would make the total trip around 9 hours, 30 minutes. You can see the link before for more details.