Move with a plan in hand.
A website transfer should account for content, DNS, email and the parts of your setup that people rely on.
First the review.
Then the move.
Map the current setup
Review the website, host, domain records, mail and external dependencies.
Prepare the destination
Copy the agreed files and database, then check the site before public cutover.
Switch and review
Coordinate the DNS change and verify key pages, HTTPS and the agreed contact flow.