If you want to obtain an SSL certificate for the new domain, please place an order for the new certificate. A CNAME record to use will be generated when you place the order.
On our hosting: nothing. Your website on the other hand may require some settings to work on the new domain. As for what to configure where, that depends entirely on your site.
For example, if you use WordPress, you need to do this:
But if you use something other than WordPress, the steps may be completely different.
If you intend to use Cloudflare, my recommendation is to set it up like this: