To wish someone well on December 31st, in English we say “Happy New Year.” If we are having an anal retentive moment and wish to use a complete sentence, we say “We wish you a Happy New Year.” (Nota bene: we usually capitalize all three words.) The Russian equivalents are «С Новым годом» and «Поздравляем вас с Новым годом». The latter literally means “We congratulate you upon the occasion of the New Year.” (Nota bene: the single Russian sound «с» is equivalent to the English phrase “upon the occasion of”. See how superior Russian is to the decaying languages of the West? Begin learning Russian immediately, if not sooner.)

The person you congratulate appears in accusative case, so if you want to add a laconic “you, too”, then the you must again use the accusative case:

At Christmastide we often supplement “Merry Christmas” with “and many happy returns!” The Russians often supplement «с Новым годом» with «с новым счастьем» “and new happiness/fortune”, which, come to think of it, is pretty similar in spirit to the English phrase, though obviously not a word-for-word translation. It is not traditional in English to add “many happy returns” to a New Year's wish, but on this occasion we’ll let that slide: