The New Year is on the 31st of December like in most countries in the world. There is a tradition to watch a solemn presidential speech 10 minutes before the New Year's Eve. Basically the president congratulates all the people of the Russian Federation with this holiday.
Russian people celebrate New-Year with their families and friends. And they stay up late, so TV channels are on with festive programs. The most popular Russian TV on New Year is the First channel or ORT as it's used to be called. Some people book restaurants to ring in the New Year.
HAPPY NEW YEAR in Russian would be С НОВЫМ ГОДОМ!
You can pronounce it as Z No'vim Go'dom. The apostrophe in between these words means that you should stress those syllables.
You can listen to an audio recording of the phrase "Happy New Year" in Russian below:
The most common greeting for this day is 'С Новым Годом - С Новым Счастьем' which means 'with the New Year - with the new happiness'. It is widely believed among Russian people that something new and exciting will definitely happen in the New Year and that the New Year will be much better than the old year.