I have different checklists for different kinds of sites, but the main idea is to copy the site as exact as possible and resolve the domain locally (meaning, update your local
hosts file to resolve your domain at the new server IP), and then update your DNS.
You will have this short window where, depending on the DNS cache, visitors will direct to one of the two copies, eventually all of them going to the new server.
You don’t need to worry about that, unless your site is down for many days. A few hours won’t change much.