Stripe will handle the subscription plans, the period (monthly, annual, etc) invoicing, the credit card security, the payment forms, discounts and pretty much everything you need to run subscription payments. It took me less than 1 day to implement using their payment forms.

There are many stripe templates on github that help you get started in almost any framework/language.

If you build your webhook correctly, you don't have to build any backend payment management, you can manage directly in the stripe dashboard and it will update your customer records directly through the webhook.