The family 「wished each other a happy New Year [exchanged New Year('s) greetings].
年賀に行く
make New Year('s) visits [calls]
年賀客a New Year visitor
年賀状a New Year's card
年賀状Nengajo are New Year's greeting cards. It is said that they came into vogue as a substitute for the personal visits people used to make to offer New Year's greetings. For this purpose, a great majority of the Japanese people now use Japan Post Service nengajo, which are issued with lottery numbers. If mailed by December 25, they will be delivered anywhere in Japan on New Year's Day. Many other people send cards of their own making, with creative touches. The most commonly used designs are based on the Chinese zodiac sign of the year. Recently e-card nengajo have also gained some popularity.(意訳:年賀状は1月1日の年始のあいさつが簡略化したものといわれる.お年玉付きの年賀はがきを使う人が圧倒的に多いが,それとは別に独自の年賀はがきを出す人も少なくない.12月25日頃までに投函すれば全国どこでも元旦に届くように配達してくれる.年賀はがきにはその年の干支えとをデザインするなど,趣向を凝らしたものが多い.最近ではEメールによる年賀状も登場している)