See greeting in the dictionary, as it says the meaning is
something friendly or polite that you say or do when you meet or welcome someone
The plural of greeting is defined as
a message that says you hope someone is well, happy, etc
In usage, to say:
is perfectly common (not to actually wish a happy birthday or a merry Christmas, but to refer to those actions). This is normal and expected from the definition.
However, the word (only plural!) can also be used in formal communication to deliver the actual greeting:
My father sends his greetings.
would be perfectly fine in a letter or even talking to someone to whom you normally speak in a more formal tone (or to whom your father speaks in a formal tone) and in this case it would be enough and you are not expected to repeat your father greetings word for word.
Greetings to you, my friends and colleagues.
is something you can use in a public speech to deliver the actual greeting (but, typically you would not do this in company of close friends; the phrase is a bit formal and it is a bit detached, but certainly acceptable in these situations described above. note: It is detached compared to other phrases simply because it lacks any specific detail, still if it is delivered sincerely it works).
If you want to be respectful and polite you can say
- Hello and welcome to my home/my office/our salon.
You can also ask how is someone doing or what kind of day they are having, and so on.