Although the whole phrase has it's own meaning, it would help to also provide analysis for each part of the phrase, i.e. gloss individual parts in addition to the whole phrase, in order to make it more memorable. I imagine a person learning English would find it easier to remember "goodbye" if they knew the individual meanings of "good" and "bye."
Yes it's "going, to return". Probably its best 1-1 translation in English is "See you later", but it's more polite than that in Japanese (even though it's considered familiar, familiar Japanese is generally a lot more polite than familiar English). The real meaning is more figurative, same as the English. "See you later" is a meaningless statement of fact, but its real meaning is more like "I'm leaving, but just so you know, I like you". Japanese of course has its counterpart, "Itterashai" which is sort of an affirmation (nice). If you say it and you don't get "itterashai" back, you can infer that the other person's feelings don't match :-/
This is a set phrase used by a person who is leaving the house.
With kanji, it looks like this: 行って来ます. The first part is the verb 行く("to go") in its て-form. The second part is the verb 来る ("to come") in its polite form. The literal translation would be something like "I will go and come back!". As a set phrase, it has a similar meaning to "I'll be back soon." or simply "I'm off."
What is the difference between the English phrases "I'm off" and "I'm off then"? Second time today that I have lost a heart for getting the English phrasing wrong but with the same meaning. The first was that I didn't know that you are not allowed to add "elementary school" after a grade.