よいお年を
This is short for よいお年をお迎えください, which means “May you greet a good new year.” This is used until December 31st (before the New Year).
あけましておめでとうございます
Congratulations for the New Year. This is the most common Happy New Year greeting in Japanese. You can drop the ございます for a less formal greeting.
今年もよろしくお願いします。
I look forward to working with you this year. Literally, this means something like “please take care of me this year also.” This is often said after あけましておめでとう to friends, family, and co-workers alike.
謹賀新年
This isn’t said, but is often seen written on 年賀状 (New Year’s cards) or on cards outside shops and homes.
迎春
This also is only written, but is often seen on store signs as a New Year’s greeting.
