I JUST learned the difference between 안녕히 계세요 and 안녕히 가세요 to say goodbye in Korean!
Use 안녕히 계세요 (annyeonghi gyeseyo) if you are leaving and the other person is staying.
It comes from the verb 계시다 (gyesida), that means “to leave”.
Use 안녕히 가세요 (annyeonghi gaseyo) if you are staying and the other person is leaving.
It comes from the verb 가시다 (gasida), that means “to go”.
So it really depends on if you are the one staying or leaving.
Those are also polite ways to say goodbye in Korean, but there are more casual ones such as:
안녕 (annyeong) - bye
잘 가요 (jal ga) - bye
내일 봐요 (naeil bwayo) - see you tomorrow
다음에 봐요 (daeume bwayo) - see you next time
Which one is new to you? Or you already know all of them? 
I like this image:
I learned more words here: