Jeff, I am not in Rhode Island but I think as far as becoming incorporated you would want to start with the following link and I would assume you need to start with the Articles of Incorporation. That should list all of the requirements and the process you need to follow. Usually you have to have a Board of Directors and a set of By-laws. But each state is different.
Most of the clubs are 501c7-at least my club in NH is. I know in NH we also had to call the IRS and get an Employee Identification Number. Something I didn't know was that EIN expires every five years unless you renew it, although when/if the new laws take effect where non-profits have to file annually with the IRS-the expiration probably won't be an issue. Previously we only had to file with the state of NH and never had to do anything with the IRS so I didn't even know our # had expired until I called a few months ago to ask a few questions.
http://www.sec.state.ri.us/corps/corpforms1205/domesticnonprofit.html