When creating the ticket initially you can choose “send email” to make Zammad send the typed content to your customer right away.
Of course, you can also solve this via triggers.
The default trigger for new created tickets should provide all the hints you need on how to solve this.
You can also send your customers trigger based notifications on every ticket update. No issue here.
As Enrike123 mentioned, please ensure that your group you’re working with has a email address configured, because otherwise it’s not possible to send your customers any mail.