There are few ways to say, depend on the situation:
Colloquial: 明けましておめでとうございます=Polite
新年、明けましておめでとうございます=Polite
明けましておめでとう!=Casual
新年おめでとう!=Casual
Writing: 謹賀新年=Formal
We say 正月(shougatsu) and 元旦(gantan) to discribe in a word in Japanese.
If you wish a good year for someone you know, you cay say:
(今年も)よいお年を! (This year) I wish you a good year for you!
If you want to have a good relationship with someone through the year, you say:
今年もよろしくお願いいたします=Polite
今年もよろしくお願いします=Polite
今年もよろしくね=Casual
Best of luck! and hope you a happy holiday!