I lived in Memphis for 25 years, but I didn't start playing golf until I was 18. I've played all things public in the area, but I have very limited experience with the private golf in town.
There aren't really any "must plays" in the public ranks that would dictate an out-of-towner should make a trip to play. Perhaps there are a few private courses that would fall in that rank if you have connections.
The public courses I enjoyed the most during my early days of golf are (no particular order):
Big Creek - As mentioned on the other thread really just the back nine is of any interest. The front nine is just back and forth on a flat piece of land and many of the holes blend in together. The back nine has the more interesting terrain. Conditioning is always questionable. I've been there when it was maintained beautifully and I've been there when it was dreadful. The courses seems to change ownership every few years. I think most recently Justin Timberlake purchased the course or at least had a stake in it.
Edmund Orgill GC - This is in Millington north of Memphis. The course sometimes hosts the qualifier for the Memphis PGA event and maybe the TN Open. I never understood how because the course always seemed to play so easy. Of course, its because I stunk at golf and I always played the white tees, which were under 6000 yards. I've broken 80 at this course more times than all others combined. It has some really fun short par 4s and it is usually in pretty decent shape for a muni. You aren't going to go home and try to impress anyone by telling them that you played this course. Nothing really special from an architectural standpoint, though I would note that the entire course is bunkerless.
Plantation GC - In northern Mississippi. This used to be one of my favorites, though it is full of stuff that would make many around here cringe (including the 2008 version of me.) House-lined, plenty of OB, really awkward par 5 first hole, and lots of other stuff. The zoysia fairways are the best thing going here - great ball-striking surface if you like grass that tees up a ball for you.
Navy base in Millington - This has a new name now as well as new greens and some re-made holes, but I'm not sure what the new name is. I know it has Eagle in the name. This is the course where I learned to play golf. My buddy took me out to the range, showed me how to swing a club, and then off to the first tee we went. I shot 131 that day, and that was only by disregarding most of the rules of the game. Over time, I came to appreciate just how difficult this course actually was. It had domed, plateau greens, may with bunker guarding the approach. There was no ground option to the greens, yet they were just firm enough to deflect the aerial attack. I'm probably over-stating this a little because I was so bad at the time. All I know is that I played it on return visits over the years as my handicap got better (not great but low teens) and I could never break 90 there. At least part was mental, but a good portion of it was those darn greens. I've been told the re-made greens are very nice and more receptive than the old ones. Don't know if that makes it better or worse any more.
Kirkwood - This course in Holly Springs, MS was designed with grand plans in mind. In reality, no one would ever drive down the 30-45 minutes to play it. It never took off like folks expected. It was probably the first time I ever played $75 to play golf. Man, did I think this course was cool. It's okay, but it is kind of quirky and again not a lot of amazing architecture going on there.
I could go on, but as I do this off the top of my head I'm coming to the conclusion that I'm just re-living good old days in my head and not actually providing any good advice for anyone else. Bottom line is that the public options are not going to blow you away.
Try to find a private course to play while you are in town. Colonial is a good option, I've heard the TPC has gotten better, I've heard good things about Memphis CC, and the Nicklaus design in Collierville (Spring something or other) is supposed to be nice.