Akemashite omedetou
Happy New Year!
The New Year is the most important time of the year in Japan. (just like Christmas in the west).
At the end of the year:
Yoi otoshi o omukae kudasai. (formal)
よいお年をお迎えください。
Yoi otoshi o! (casual)
よいお年を！
During New Year's days (Jan.1st to 3rd), up to the middle of January:
Akemashite omedetou gozaimasu. (formal)
あけましておめでとうございます。
Akemashite omedetou. (casual)
あけましておめでとう。 "Akemasu" literally means "to open". "Kotoshi mo yoroshiku onegaishimasu (I look forward to our continued relationship over this year)" is often added after "Akemashite omedetou gozaimasu". To reply, "Kochira koso" is used.
Seeing someone after a long absence
Gobusata shite imasu. (very formal)
ご無沙汰しています。
Ohisashiburi desu. (formal)
お久しぶりです。
Hisashiburi! (casual)
久しぶり！
There is a Japanese song titled "Ohisashiburi ne（お久しぶりね）". "Ne（ね）" is a sentence particle. Ne is used to seek confirmation and is similar to English expressions such as "right?" or "don't you agree?".
To reply to "Gobusata shite imasu," "Kochira koso (Same here)" is used. In casual conversations among friends, simply repeat "Hisashiburi!" or "Hisashiburi ne".