送舊迎新 adieu, old year; ring in the new year

送 (send) 舊 (old) 迎 (greet) 新 (new)

Happy New Year. 新年快樂.
(simplified, 新年快乐)

新 (new) 年 (year) 快 (quick) 樂 (cheer)

Also,

謹賀新年
謹 (prudent) 賀 (congratulate) 新 (new) 年 (year)

Yes. Prudent New Year to you, too.