If you choose to host your web site through lower end hosting option or with a third party shared hosting provider we/they can offer email as part of that package. We strongly advise however that you do not used shared hosting domain based email. This is because on shared hosting everyone shares a connection and the actions of one can, and do, reflect negatively on the other people sharing that system in the eyes of the systems deciding if your emails are spam.
We recommend our clients use GSuite, the business services system offered by Google. GSuite allows you to have the usual Google services like Gmail, Calendar, Drive, Docs, YouTube etc. but using your own domain. So in our case we use the differo.co.uk domain both on this website and on GSuite allowing us to have our emails handled by Google and share calendars and files between staff.
We are happy to set you up on GSuite and do not take a fee from you for doing so. That fee is paid to us by Google. GSuite costs £4.60 per user per month, paid directly to Google.