Can those of you used to competitive target shooting tell me how to go about finding a fun target shooting group? I've heard rumors about groups that are for fun, shooting balloons, gallon jugs of colored water, and similar such things. I would like to join something like that.

Look for groups which shoot steel plates or Steel Challenge. Start with the NRA website for clubs in your area and then check out their individual websites for details.

Another vote for steel shooting.

Most clubs only allow steel or paper products for targets on a range.
Watching the "Top Gun" shows shooting all that glass and plastic makes me wonder how many people they have to clean up the mess on the range. Certainly no legitimate sanctioned pistol competition allows all that.

Based on the experience at our range, the people who bring in all the prohibited "fun stuff" to shoot just leave the incredible mess and sneak away.

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Science plus Art

cowboy action shooting is all about the fun, i was surprised at how friendly and helpful EVERYONE was/is.
or you could just make your own group, invite some guys from work ,take some kids out for their first time ,maybe teach a girl how to shoot .you cant help but have fun.
x2 on the range slobs...whatever you shoot make sure you can pack it away when you are done

I'm sorry that I didn't answer your suggestions right away--I'll try to answer everyone now.

I hadn't thought about the mess cleaning up an outdoor range. The group that I heard about that shot balloons and similar things must use an indoor range.

Perhaps I could get a group going. However, since I am a "newbie" myself, I would have to get an experienced range master to make sure that we followed safety procedures.

My instructor taught me to clean up the range. Actually, I find sweeping up [but only behind the shooters] gives me something to do when I'm waiting my turn on the range. People who leave a mess should lose their range permits, I think. I for one don't want to slide on empty shells and fall into a shooter!

I'll check out a cowboy action shooting. If they let me bring the 617, I'd go for it.

617 wont fly in CAS. youll need a couple single action revolvers and a s/s shotgun or pump with exposed hammers also a 10 shot lever or pump in pistol caliber also with exposed hammers.
gives you an excuse to buy more guns ,even if you dont have all your equiptment somebody will probably lend you what you need .i shot my first two matches with other peoples rifles ! its no big deal .people are super friendly ...not like those wannabes and their bottom feeder clubs lol (j.k.)

Check around at your local gun clubs. Ours has random matches once a month from indoor targets to outdoor rotten-egg shoots among others.