If you're using Heartbeat Payments, you can set up notifications for payment related activities in your Stripe Dashboard. 

To access these settings, go to Settings > Billing > Subscriptions and emails or search for "Subscriptions and emails".

These settings will allow you to receive email notifications to your Stripe account email for every successful payment. You may also consider toggling ON dispute and fraudulent payment email notifications here.

Stripe does not have a core setting to send email notifications for new customers. However, you can use automation platforms, like Zapier, to set up further notifications as is covered on the Stripe Zapier page.