Köszönöm is the first person, singular of the verb köszön, which means "thank".
For a long time, I only said "viszlát" because I couldn't remember the whole thing. You can break it down into: viszont ("again", like the word "vissza" which means "back"), látás ("seeing", like "lát", which means "see"), ra (a case suffix; don't worry about this yet).
I found a good way that helps me. I took the time to write on paper each Duolingo example (just the Hungarian). It's so different from English (my native language) that it seems the only thing to do is to flat-out memorize things. I look at what I've written from time to time during the day.
I think it is better to deconstruct the words and sentences early on to what every part of means/how every part plays a role to construct the meaning in the sentence, just like how david.altmayer (the post above you) did it. Once you're done from that, try to compare the learned deconstruction that are the basic things you need to memorize in order to use the language. Basically maximize learning relationships and basic patterns over memorizing whole chunks of words at once.
(edit : I deleted a large section of this comment just to reduce the size and remain in context xD)
Comments here are helpful. This is the first time I am reading posts. I've been thrown by the punctuation being "off" to my sense of grammar. Does anyone know if this is the proper punctuation for Hungarian? -- Meanwhile, I will take courage that I will get the spelling and parse the wording right!
Viszont - likewise, each other
lát - to see
látás - the seeing
látásra - onto the seeing
Viszontlátásra - (on)to seeing each other (again)
It's a bit convoluted since the grammars used in Hungarian and English are very different, but breaking down words helps a lot with saving memory space.
Maybe somebody will correct me, but "see you" in english is the informal "goodbye", at least I know I would only use "see you" for friends.
"viszontlátásra" is a formal greeting, I would not use that with friends.
In this way I could argue that "viszontlátásra" is best translated as "goodbye".
You are right about the English "See you." It is short for "See you later" or "I'll see you later"---or anything else like that. It is a bit like the German "Auf wiedersehen," but you are right that "See you" is only casual language, for friends. "Goodbye" is suitable for casual or formal occasions.