I don't yet know the scope of everything I might need, but right now I believe that a customer management system is the focus. A way to input their details and payment information to have an automatic recurring membership.
I don't think I really need a way for them to view their account as it won't have any fancy features. I just need to know that they are a member and have a membership renewal. They can sign up and cancel in person or over the phone (I'm not making it disgustingly hard for people to cancel like the large gyms).
On top of that I would imagine it would be nice for the software to keep track of my total number of patrons and thus a total income amount