There are four ways you can do this:
1) You can setup your very own email server on your very own VPS. There are plenty of tutorials online, but this is not an easy task. I can do it but I don't think I would want to be in that position of troubleshooting when things go wrong.
2) You can use one of the many available web hosting control panels. Many of them come with mail settings included, one of which is Sentora – https://www.sentora.org/
3) You can simply choose to pay. Gmail, Outlook and Zoho mail. There are plenty more services out there. I am not allowed to promote anyone, but Zoho offers up to 25 emails free when signing up. You only pay for more domains or more space... or more than 25 user emails.
4) You can leave your emails on the shared hosting platform and simply point your DNS / MX records back to the shared hosting. Same platform, not running around backing up and emails. The setup is also familiar to your clients.
Conclusion: It's really your call. What do you feel comfortable with? And sometimes, it's good to give your clients the options. The might just pay for premium emails. Educate your clients – first rule of business. Educating your clients is key!