Craig,
I'm a member at Legends Club, which is somewhat unique in that it has two eighteen hole courses, one of which is private and one public. Both were designed by Bob Cupp with Tom Kite getting credit also. Please see my instant message to you.
Nashville has surprisingly little, if any, classic architecture. Belle Meade is a tremendous Ross routing, but little work of his remains after two renovations. I would place it and Fazio's Golf Club of Tennessee at the top of the list. Other notable privates include Richland CC, which relocated from the in-town Ross design that hosted the Women's Open aorund 1980 to a hilly Nicklaus layout in the suburbs. Palmer's Governor's Club is rather bland, but a course of surprising interest is Nashville Golf & Athletic Club, a monstrous Devlin/Von Hagge 70's layout with a unique set of greens. The course was so long at inception that the tips are presently located at the front of several long runway tees. Hillwood CC is a solid, traditional layout in a parkland setting without any significant architectural lineage.
On the public side, three courses stand out: Legends Club Roper's Knob (the LPGA event has been held on the private course), Springhouse - a Larry Nelson design affiliated with Opryland Hotel that hosts a senior tour event, and Hermitage Golf Course, which has two courses, one of which hosted the LPGA tour for years. I believe the newer course is better and is a Denis Griffith design. The older course was designed by Gary Roger Baird.
Hope this helps. Of interesting note, there is a new nine-hole course west of town built by a farmer who never played the game. I haven't seen it but must get out there soon.
Regards,
Mike