My first winter experience was in South Korea in November 2015. It wasn’t even snowing but the cold was biting. Winter is bearable when there is no wind but apparently winter months in South Korea have the strongest wind (source).
Your best bet to experience snow is to visit PyeongChang in South Korea from February 9 -25, 2018. They will be hosting the 2018 Winter Olympics. I know Japan is also beautiful at this time of the year but I have not been to there and I am hesitant to apply for a visa. I guess I am not ready for Japan yet.
Here are my top picks for winter destination. Check number 6!