For best experience, add the new domain as an email alias (I suppose you want to make it primary) and keep the old one as secondary. Both PowerShell and email address policies (on-premises) can help you do that.
As long as you do not change the UPN, the login experience should not be affected. If you have Entra ID integrated apps that depend on primary SMTP address, you will have to update their configuration.
Similarly, if you plan to update more than just he primary SMTP address attribute, you might have to perform additional actions with third-party apps. Within the M365 suite, UPN and the primary SMTP address are your only concerns.