A lot depends on what you expect from competing, and also on what the local club is like. There is an ICORE club in OKC, for example, that will tell you right up front you need an 8 shot 9mm (929) to be competitive in their matches: they aren't interested in 6 shot friendly matches. Since I only shoot 6 shot classic revolvers, I just don't go to their club anymore.
Other ICORE clubs do Classic matches.
IDPA is a good place to start with a 686 or 67, since IDPA is 6 shot only, 4" (nominal) barrel. Revolvers are scarce at some clubs just because SSP and CCP are most popular. IDPA has no more rules than USPSA or ICORE, and if anyone has not read the new rules lately, they don't even know what IDPA rules currently are. I actually prefer the previous IDPA rules, where I could shoot on the move, and taking cover actually meant getting behind something, not just standing at a fault line.
IDPA now allows more "run and gun" and we who have been in IDPA over 20 years call the new rules "Son of IPSC." But nothing in the rules REQUIRES you to try to shoot at the National level of intensity and speed.
I normally shoot Production Division in USPSA with a semi-auto because with my Model 66 shooting minor 6-shot, I am not going to win anything in USPSA Revolver Division. I have even had one uninformed RO tell me I had to have a major caliber to shoot 6-shot in USPSA, which is absolutely not true, you can always shoot just for fun.
Steel challenge is the easiest competition to start because you just stand and shoot rather than dealing with complicated movement. You can even use a .22, but don't be surprised if a 12 year old girl beats you.
USPSA/CRO
IDPA/SO (previous IDPA Match Director for 15 yrs)
P.S. NRA Action Pistol is my favorite, but I haven't been close to any clubs sine I left KY decades ago. Par time scoring and accuracy is King!