Connecting new domains

If my client wants to purchase new domains to connect to his existing site (he wants a couple of domains and emails for one website) how do I do this?
Do I contact the hosting company?

One can redirect one domain to another one. You can do that in the dashboard of your hosting account.

Yes – there are redirects and there’s setting domains as an alias of the primary domain, which accomplishes the same thing. Regardless, hosting company can help.