Which one is correct?
Happy new year to both of you.
Happy new year to you both.
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Both are fine, and there's really little difference between them. Other examples:
I'd like to take both of you to dinner to thank you for your help
I'd like to take you both to dinner to thank you for your help.
She gave them both her blessing.
She gave both of them her blessing.
I bid you both adieu.
I bid both of you adieu.
In some cases one sounds better than the other, but I think this is often what is familiar and preferred rather than related to any grammar rule.
Although both are grammatically correct, if you check the Cambridge definition for the word both, you will know that both is preferably placed at the beginning or in the middle of the sentence. So it leads me to think that placing both between to and of makes the sentence sound more proper.
Happy new year to both of you
sounds much better and more suitable.