The faster method.
WHY BOTHER
Faster than any web interface, and it works without an internet search.
Available on every operating system.
WHAT TO LOOK UP
The domain's A record, for where it points MX records, for mail TXT records, for authentication NS records, for which nameservers answer
QUERYING A SPECIFIC NAMESERVER
The most useful capability: ask your authoritative nameservers directly rather than a cached resolver.
Most command line tools accept a nameserver as an argument.
That is the check that distinguishes a change not made from one propagating.
WHAT TO LEARN
Two or three commands cover everything most people need.
The syntax differs between operating systems; the concepts are identical.
WHAT TO WATCH FOR
Which nameserver answered Whether the answer came from cache or authoritatively The TTL remaining, which tells you how long until a cached answer expires
THAT LAST DETAIL
The remaining TTL on a cached answer tells you exactly how long before that resolver asks again.
Useful when explaining to someone why they still see the old site.