Before, during and after.
BEFORE
Record or export the current zone Note which provider holds DNS Note the TTL on records you will change Lower TTLs a day in advance if you can
Note what must not change: mail records, verification records
DURING
Make the change in the correct place Confirm it saved Query your authoritative nameservers directly to confirm the new value
AFTER
Test the website Test mail, both directions Check every subdomain Check anything relying on a TXT record Run AutoSSL if the address changed
WITHIN A WEEK
Confirm nothing was missed Raise TTLs back Update your documentation and your zone export
FOR A NAMESERVER CHANGE
Every record must exist at the new provider before switching Verify by querying the new nameservers directly Do not cancel anything until propagation is complete
THE TWO THAT CAUSE MOST PROBLEMS
Mail records not recreated, and changes made in the wrong place.
Check both before concluding anything.