Been there, done that.
My personal method is to buy the new domain, initially add it to my current web hosting as a Parked Domain, change the domain name in WordPress (see info below) and work with my web hosting company to make the new domain the main domain and the old domain the Parked Domain.
If you don’t plan to keep the old domain forever, you will, at some point, want to notify people who use the old domain that it has been changed and that they should change their bookmarks, then let the old domain expire after 6-12 months.
The only piece missing is to be sure any sites that link to you have also updated their URL to link to you.
I would recommend reading the relevant sections of both of these:
https://codex.wordpress.org/Changing_The_Site_URL
https://codex.wordpress.org/Moving_WordPress
The key part of all that is to be sure that “hard coded” URLs on your site get changed, which generally involves Search and Replace operations on the WordPress Database.
