As a newly minted Memphis resident, there's a lot here, but it's so spread out. I also haven't played any Nashville courses, but if the reviews and media here are anything to go by, Nashville is probably the better of the two.
Having said that, here's a quick rundown of the Memphis area courses (I haven't reviewed any because I want to play them a few times before I review each one):
Central Park in Southaven, MS, is probably my favorite course in the area of the ones I've played so far. Fun park course with a good mix of different throws. Few long holes, few short holes. Lots of tweener par 3.5's that are all fun to throw. Decently easy to navigate too.
MAM DGC. There are pictures of all 18 holes over at discgolfscene. Haven't played it yet but heard it's a fun course.
Shelby Farms. Make sure you're using Udisc, as there aren't tee signs right now while the club finalizes the new layout. There are a few signs still hanging on, but the key thing is to go up and to the right in the woods after hole 2 to get to the new tee for hole 3. Additionally, the tees for 16 and 17 are dirt right now, and the marking for 17 is really lacking. It's also a mostly wide open bomber course, although there are a few shorter holes and some technical-ish greens. Fun to throw.
Libertyland. Go during the day, it's in a rougher part of town. Central BBQ, Memphis Made Brewing Co., and the Railgarten bar are right near it though. I didn't finish playing it when I went because my girlfriend was with me, she wasn't a fan of the area, and the only holes I had left were wide open field holes. If you can deal with that though, there are some extremely fun holes using the spots of mature trees. Sadly, to make a full 18 they had to have a few throw in the field type holes, but it's alright.
Johnson Road Park (JRP) is a fun park course. First five holes are seriously wooded, then the rest is parkland. A few holes where you can lose a disc in the ponds so be careful, but otherwise super fun course to throw.
Sycamore Creek DGC at Hinton Park: the name is longer than the course, but it's a nice little nine hole course. Couple really silly holes but some solid ones as well.
To the north, there's the two private Bud Hill courses, which I've heard are great, and the two courses at Meeman-Shelby state forest, which are heavily wooded but I've heard are fun as well. Out towards the boondocks you have Edmund Orgill, which is really tough to navigate but fun to play. I think Udisc has the original 18 hole layout in the maps. To play the 20 hole layout, after hole 7, walk across the road and there should be a tee sign. Hole 8 is the pad near 5's raised basket back across the road. Then, hole 20 is teeing from under the big trees across the road from 19's basket down to the practice basket. Also, when you get to the park and enter it, the disc golf parking is in the same lot to the left as the boat parking. There are also a pair of courses in Southaven or Horn Lake MS at Latimer Park. I haven't played them but the locals there play them all. the. time.
Additionally, Memphis has a lot of other cool stuff. The Rendevous downtown makes amazing pork ribs, and all 5 breweries in town make excellent beer. Wiseacre doesn't have a huge selection but they make great beer, and the outdoor space is really nice.
If you pick Memphis, hit me up, if I can I'd love to throw a round!