After successful transactions Charge can trigger actions. These actions can send emails, and update entries. Actions can also add members to subscriptions, which control a member's member group based on their payments.
Actions are simple to setup, and just require an extra parameter on your payment form to use. Check out how to use Actions.
Subscriptions let you control a member's member group based on their one off and recurring payments. To fully use subscriptions with recurring payments you'll also need to setup Stripe's Webhooks to callback to your site.