Scratch

First of all you would need to get permission from your local library to download scratch onto the computers.
Once you get permisson put up poster at the library telling people about the club. Put your e-mail on the poster so people can get further details for the club.
That's about it hopefully that will get things started.
(The chances of getting permission are very high) Goodluck!

You are gonna need at least two turntables, a PA system, A microphone... just kidding 

The library itself should have a lot of ideas about hosting any kind of club, ask them. You really need an older tech-type person(s) familiar with Scratch who can oversee the startup of the club as well as drop in occasionally, in my opinion.

And we can help, send them to this forum when they have scripting questions.

I organized one of these for about a month, and it was always very helpful for me to have at least one or two members who were very good at scratch, to answer the flood of questions from the rest of the group.

Thanks for the advice offered so far.  You've all been generous.  I will use it all.  Problem is this:  where do I find another adult experienced with Scratch to help out?
I have tried to learn Scratch, but I haven't had time.  My son is quite good with Scratch- but he's only ten.

Would it be possible for me to be there as an adult advisor, and then to have just a few kids, like my son who are good at Scratch, who can be resources for kids who are new to Scratch?

If you live anywhere near boston, the Scratch Team is built up of MIT students, so you could see about getting one of them. If you live somwhere more than an hour away, I suggest asking around on the forums or using those resorces.