First decide how the move will happen: a host's own migration plugin or service, or a manual transfer of files and database. The guides on each host's migration tools explain what they document as covered and what is left to you, so read the one for your destination host before you schedule anything.
Next, check the parts of the site that sit outside WordPress itself: nameservers and DNSSEC, email routing, payment webhooks and plugin licenses tied to a domain. These are the items most likely to fail quietly, after the site looks fine in a browser, so confirm each one before and after the switch.
If something already broke, go straight to the fix-it guides for the exact error you see, such as a database connection error, a critical error message or mixed content warnings. Each follows the documented troubleshooting steps in order, and each assumes you have a current backup before changing anything.