Choosing a template & CMS for service selling website

Hello !
I haven't been working with that in the past, so I find myself pretty much a total newbie in the present situation, it's refreshing, haha. I'm coming to ask for CMS suggestions, please

I'm working on a project of a business, in which we would be selling a service to the visitors. They pay us, and every month, we do stuff for them, in their stead.

For this, we need a website that would be
- simple
- convincing
- and that would include a membership and payment system (by credit card, or various e-wallets)

The first two elements, simple and convincing, rely on choosing a good design.
But the third option is more complex, for me at least. I have no idea how to integrate, into a website (either a static website based on simple php and html, or a wordpress basical blog if it works better for that purpose)
> a payment system
> a customer membership system

I already integrated payment solutions into wordpress and oscommerce for instance, but here, this is different, there is no predefined ecommerce engine, and we don't want a complex powerhouse in which customers have baskets and the like.
Basically, I need to allow visitors to become customers, and then us, the staff, we need to be able to have a list of who is currently a customer, with their names and contact info. Just that.

Would you know how that can be easily achieved ?
Sorry to sound newbish, but I AM a newbie for this

Thank you very much if you know how to achieve it, I'll be grateful
Kind regards,
Sabin