Ahh, well then I'd imagine IDPA or maybe USPSA will be your best fit.
You can go to either organization's web sites and use their "Club Finder" to locate who in your area puts on regular matches. Then call their contact number and see if you can find out when they do their practice nights/sessions.
I'm the Match Director for IDPA at a club in PA. We get LOTS of new shooters, many of whom have never done any competitive shooting at all. (Some who've never fired a gun before!) We always encourage folks to come out to our weekly practice (Thursday nights, in our case) and we'll give hours of one-on-one and group informal instruction in both shooting, and in the specific discipline -- absolutely free! We can also usually get someone some trigger time on several different kinds of guns to help them figure out what they'd most like to shoot in competition.
You'll probably be welcome to come out and visit any affiliated club on a match day to see how it all goes, but I can promise that the MD and his Safety Officers are going to be busy and/or pretty worn out that day, and probably won't have a lot of time to give you. But find out when they practice and I'm sure you'll be welcomed with open arms!