There are a lot of things to consider, and some of these items have been touched on already.
As mentioned, liability is a big concern. If you are organizing rides, you can be held personally liable if someone is injured on the ride. There are a bunch of scenarios in which a lawyer could file a suit against you as the ride organizer, and while you may ultimately not be found liable by a court, the legal fees alone can be quite expensive. And yes, it has happened that people have sued others on group rides and charity rides.
For that reason, it can be worthwhile to incorporate as a club to put any liability on the club, and not you as an individual. To do this, you likely need to file incorporation papers with your state creating the club and laying out all your corporation's governance - board members, meeting requirements....
Once you are incorporated, you can then consider becoming a 501(c)3. You will need to work with the IRS to justify the nonprofit status. The application itself isn't difficult, but the follow up questionnaire's can be a bit nerve rattling. If your organization's annual revenue is less than $25k, you only need to file a postcard return each year attesting to your revenue. If it is more than $25k, I believe you have to file a full 990, which can be complicated and may necessitate and attorney.
After the club is official, the next consideration is if you want to be USAC sanctioned. It's not terribly expensive, and it does give you some benefits.
The first is the opportunity to purchase liability insurance. We found that the general liability policies through USAC were much cheaper than others. I believe we also get our officers and directors insurance through it as well. Both policies are good to have in place, and they will help defer legal costs should the organization be sued for any reason.
The second is that should you opt to hold an official event, such as a charity ride or race, you can have it sanctioned through USAC, which also gives you access to event insurance. Any time I have spoken with a property owner about hosting a cycling event at their property, they have asked about insurance. The USAC event insurance is pretty standard and solid.
You also have the option of charging dues. Depending on the size of your club, dues can be as little as $25-$35 a year. For our club, that generates enough revenue to cover the annual USAC fee and general liability and officers and directors insurance policies.
That's the quick summary of the process. I am pretty sure USAC has information on its site for how to go about forming a club.