I would say something like:
'I wish you all the best for the New Year' - "Бажаю тобі/Вам всього найкращого в Новому році!"
I'm not sure whether "Новий рік" (New Year) should be capitalized in this case, but it is when you are talking about a holiday as such. You should use "ти" (Spanish "tú") if you are familiar with the person and "Ви" (Usted) when in formal situations, with elder people, etc.
Oh, and when you refer to two and more people, you should use "вам" in all cases (not capitalised).
P.S. Welcome to the forum, Swotty
