Not quite. To understand better, it might help to break it down.
What most people use as "Hello" in Korean, in the polite, but not formal speech is Annyeong Haseyo? (안녕 하세요?). Annyoeng literally means "Peace," and this appears in what you're hearing for hello and goodbye both I'm guessing. Sometimes, this is used by itself for hello or goodbye, but that is usually only with children. The second word is "Haseyo" which means "do" are "are you doing" or "are you" in this case.
There are two expressions for goodbye that use a derivitive of Annyoeng, and if you're not very fmiliar with the sound of the language, you have to listen carefully to hear the difference. If the person you are parting company with is going someplace, you would say "Annyeong-hi gaseyo." The hi (hee) added onto "Anyeyong" changes it from an adjective to an adverb, and is necessary if you are going to use it when telling someone to go this way. "Gaseyo" (with a g sound as opposed to the h sound in the hello greeting) means "go." So when that person is leaving, you are telling them to go peacefully: 안녕히 가세요.
If the person is not going anywhere, you would say "Annyeong-hi kyeseyo." "Kyeseyo" (계세요)(kyeh-seh-yo) is the honorific form of isseyo (있에요). It means be, exist, stay, all of the meanings of isseyo. You use this in departing when the other person is not going anywhere. This would include when you leave that person's house, when you end a skype call with them, when you depart on the street and that person is not going anywhere, etc.
There are other goodbyes for things like when you are departing someone's business establishment, or if you are telling the person to have a good day, etc. But I think the question you were asking was about the three I wrote about. Hopefully, this clarifies it. If not, please ask for clarification.