There are several grottos in your area. Chattanooga meets the second Monday, Pigeon Mountain Grotto meets the first Friday and the Sewanee Mountain Grotto meets the second Saturday. There are great cavers in all of these grottos that you could learn from. Also, each grotto offers monthly caving trips.
Chattanooga meets at 7pm at the Chattanooga Aboretum & Nature Center (400 Garden Road Chattanooga, TN 37419).
http://www.facebook.com/#!/groups/137404042945040/Pigeon Mountain meets at the Bank of LaFayette at the Rock Springs branch on Hwy 27.
http://www.facebook.com/#!/groups/55499438417/The Sewanee Mountain Grotto rotates their meetings. They are typically held at a grotto members house on Old Sewanee Road, Sewanee, TN (not far from Kimball, TN).
http://www.facebook.com/#!/groups/144509055567142/.
Have you joined the National Speleological Society (NSS)?
www.caves.org.
There are also several great caving events held in the area each year.
The Southeastern Regional Association of Cavers (SERA) has an event each year that is hosted by individual grottos. It is called the SERA Summer Cave Carnival - this years event was in May.
http://www.caves.org/region/sera/CaveCarn.htm.
The Sewanee Mountain Grotto hosts an event each year called Cave Fest. It is held Labor Day weekend.
http://www.facebook.com/#!/events/299738240109431/And the Dogwood City Grotto hosts an event called the TAG Fall Cave In. This year it will be October 4-7, however you must be an NSS member or come as a guest of one to attend this event.
http://www.tagfallcavein.org/.