What kind of competition?
Steel City (Hoover) shoots IDPA
http://www.pistolpro.com/ssidpa/Monthly.shtml
also SASS, Highpower (rifle), and a Practical match with no affiliation
http://www.scssa.org/
Heritage (Argo) shoots IDPA, USPSA/IPSC, and SASS.
http://hrpclub.org
as does Moss Branch (Anniston)
http://www.mossbranch.org
F.O.P. Range Inc. (Pleasant Grove) shoots USPSA/IPSC
http://www.scssa.org/
There is a range in or near Birmingham that shoots NRA Conventional Pistol (bullseye) but I don't know where. I dug through the NRA site and found a contact at:
ALABAMA, Birmingham: 2700, Apr. 3, May 1, Jun. 5, Aug. 7, Oct. 2, A; L.W. Meyer, Jr., 5701 Ave. P., Birmingham, AL 35228
Go to any of these ranges and you will get to shoot and meet people who can direct you to other ranges within a reasonable distance. I live in Florence and shoot IDPA in Sheffield, Huntsville, Arab, Hartselle, and Birmingham; and will be going to that Cystic Fibrosis Benefit shoot in Anniston at the end of April.