Nashville & Clarksville
I don't know anything about Clarksville.
A barbecue place that I have really been enjoying lately in Nashville is Jimmy Carl's Lunch Box. It's operating out of the Station Inn, which is a tiny music venue in the "Gulch" neighborhood near downtown. Ribs are outstanding. Pulled pork is very good as well. And be sure to "try the beans." Here's the thing, though - they're only open Tuesday through Friday, and only for lunch.
If you want to stray a little further from Nashville (in the opposite direction of Clarksville), Martin's Bar-B-Q Joint in Nolensville, Tennessee is pretty much the consensus favorite as best barbecue in the Greater Nashville area these days.
The easiest route would be to just go to Jack's BBQ in downtown Nashville. It's right in the tourist zone and it's open for lunch and dinner. And although it's not the best barbecue in town, it's not bad.
There are many great options for southern food in Nashville. Arnold's is my favorite. It's essentially downtown, but again, it's only lunch M-F. The Loveless Cafe is good, as long as you stick to country ham and biscuits, but it's a bit of drive from downtown. Monell's is fantastic, too. You sit at big tables with strangers and are served family style, all-u-can-eat. The food is overall very good. Other places that will not disappoint: Swett's, At the Table, and Dandgure's.