あけましておめでとうございます。
Akemashite omedetō gozaimasu.
Best wishes on the beginning of a New Year.

This is the standard expression for greeting people for the first time in the New Year, the celebration of which is the most significant and ceremonially observed holiday period in the Japanese calendar. A complete formal New Year’s greeting begins with these words and continues with an expression of gratitude for the other per- son’s kindness (or patronage) over the course of the previous year:

旧年中はお世話になりました。
Kyūnenchū wa osewa ni narimashita.
I appreciate your kindness throughout the past year.

The speaker then expresses the hope that the New Year will see the continuation of good relations, communicating this desire in the form of a request:

本年も相変わりませず、よろしくお願いいたします。
Honnen mo aikawarimasezu, yoroshiku onegai itashimasu.
I humbly ask your continued favor in the coming year.

Even such a formal greeting, however, will have the hollow ring of a memorized formula if your delivery is rushed or if the words are tossed off in a monotone. It is essential to deliver your greeting at a measured pace, be attentive to the reply, and bow once respectfully upon concluding the greeting.

Among friends, from whom an extended and formal New Year’s greeting would sound stilted and unnatural, a shorter and less formal alternative is preferred:

明けましておめでとう。今年もどうぞよろしく。
Akemashite omedetō. Kotoshi mo dōzo yoroshiku.
Happy New Year! Don’t forget me this year!

While the general trend is toward less formality, this is an excellent opportunity to represent yourself as a solid, serious person to people in your neighborhood, for example, or to higher-ups at the office, by extending a scrupulously proper formal New Year’s greeting, as described above. These few phrases, skillfully executed, could actually help to restore a less than impeccable public image, if need be. After all, New Year’s time is the season for making fresh starts. The key to making a good impression is the delivery: enunciate each word clearly to the very end of the greeting, finishing with a crisp onegai itashimasu. The big finish is a sure crowd-pleaser.

– Source: A handbook of common Japanese phrases –