My name is Lilith, I'm on the Social Care team at Shopify.
Happy to hear about your new domain! It’s always exciting to own a brand new domain.
Shopify does offer you the ability to purchase your own custom domain that you can manage in your account admin. You can certainly use your domain to create your own branded email addresses, but our e-commerce platform doesn't include a built-in email service similar to Gmail. We like to recommend other specialized email service companies for this type of function.
On your admin (Online Store > Domains) you're able to create forwarding email addresses. This means you can create custom addresses that you can set up on your account under Customer Email to use for your customer notifications.
Having these custom branded emails help create a cohesive brand for your business. Any email replies to these custom forwarding email addresses will be sent to an existing email address that you have an inbox for. You'll not be able to reply from these custom addresses as you won't have an inbox for them, that is why they are known as forwarding email addresses.
If you do want to create an inbox for your custom email addresses where you can receive and send emails, you'll want to create an account with G Suite or Zoho. I've had a chance to use G Suite on several businesses that I've built and found that this is one of the most affordable and easiest options to work with. It's also pretty painless to sign up and integrate with your new domain on Shopify. You'll also notice that a lot of large companies these days are also moving their email hosting to Google. They know email!
With G Suite you'll have a pricing plan to sign up with, I'd recommend sticking with the lowest priced plan since you can create nearly limitless email aliases on your new domain and you can send an email with any of those aliases. Normally a set of "info", "support" and "newsletter" are common aliases that are good to start with to manage your business.
Let me know if that helped! I'm happy to answer any questions you think of as you work on your business.