Quick answer
The big one is Но́вый год, not Christmas. Congratulations always start with С… plus the instrumental: С Но́вым го́дом!
The main holidays
Но́вый годnóvy godNew Year — the biggest
Рождество́rozhdestvóChristmas — 7 January
8 Ма́ртаvosmóye mártaInternational Women’s Day
9 Ма́яdevyátoye máyaVictory Day
Ма́сленицаmáslenitsapancake week before Lent
Words
пра́здникprázdnikholiday, celebration
пода́рокpodárokpresent
ёлкаyólkaNew Year tree
Дед Моро́зded morózFather Frost
фейерве́ркfeyervérkfireworks
вы́ходнойvykhodnóyday off
What you say
С пра́здником!s prázdnikomHappy holiday!
С Но́вым го́дом!s nóvym gódomHappy New Year!
Поздравля́ю!pozdravlyáyuCongratulations!
Жела́ю сча́стья!zheláyu cháastyaI wish you happiness!
New Year is the big one
Christmas was suppressed for seventy Soviet years, so the tree, the presents and Father Frost all moved to Но́вый год. Families eat оливье́, watch the president at midnight and then the same film as every year.
С + instrumental
Every congratulation follows the same pattern: С Но́вым го́дом, С днём рожде́ния, С пра́здником — “with the …”. Learn the pattern once and every holiday is covered.
Typical mistakes
С Но́вый год!
С Но́вым го́дом!
с takes the instrumental.
Congratulating a woman on 23 February
8 Ма́рта is for women, 23 Февраля́ for men
The two days mirror each other.
Check yourself
Which is correct?
с + instrumental.