Quick answer
С Но́вым го́дом!
s nóvym gódom — the biggest greeting of the Russian year
The options
С Но́вым го́дом!Happy New Year!
С наступа́ющим!Happy New Year — said *before* midnight
С Но́вым го́дом, с но́вым сча́стьем!Happy New Year and new happiness!
Жела́ю здоро́вья и сча́стья!I wish you health and happiness!
Before or after midnight
Before the year turns you say С наступа́ющим! — “with the approaching one”. After midnight it becomes С Но́вым го́дом!. Getting this right is a small but noticed detail.
What happens at midnight
The president speaks, the Kremlin clock strikes, everyone drinks шампа́нское, and many write a wish on paper, burn it, and drink the ashes in the glass. Yes, really.
What you hear back
И вас та́кже!You too!
Спаси́бо, и вас!Thanks, you too!
Ура́!Hooray!
Typical mistakes
С Но́вым Го́дом with capitals
С Но́вым го́дом
Only the first word is capitalised.
С Но́вым го́дом on 29 December
С наступа́ющим!
Before midnight it is “the approaching one”.
Check yourself
It is 30 December. What do you say?
Before midnight: С наступа́ющим!