Moving where a site is served from.
WHAT TO CHANGE
The A record for the domain, and for www, to the new address.
Any subdomain A records pointing at the old server.
WHAT NOT TO FORGET
Mail. If mail stays where it was, the MX records and the mail hostname's A record must continue pointing there.
Changing every record to the new server moves mail too, which may not be what you want.
THE COMMON MISTAKE
Updating the website records and forgetting a subdomain, which continues serving from the old server.
Visitors see an old version and nobody understands why.
BEFORE CHANGING
Lower the TTL a day in advance, so the change propagates quickly.
Note every record so you know what you are changing.
TESTING BEFORE THE CHANGE
A hosts file entry points your own machine at the new server while everyone else still reaches the old one.
Test fully before changing DNS.
AFTER CHANGING
Confirm every hostname resolves to the new address.
Check the site, mail, and any subdomain.
KEEP THE OLD SERVER
Until propagation is complete and everything is verified. A week is sensible.