Translation:Well then, I'm leaving.
The expression is used as a whole phrase containing both of those meanings. It means that you are leaving a place that you expect to return to, usually before too long, such as home or your workplace. In English, we are more inclined to say just one of the two than both, for example: I'm leaving or I'll be back soon.
You are quite right about the literal meaning. So, an alternative to Well (/So, /Ok), I'm leaving OR ... I'm off, might be: I'll be back. Granted that we might be more likely to say that if we say "soon," "later," or specify a time. I will return, on the other hand, may sound a little too much like General MacArthur.
The て form can be used in a lot of different situations. In this case the structure is "verb in て form + くる". It can be used to express that you go somewhere to do something (depending on the first verb) and then return back to the place you where at when you said the sentence.
行ってくる is unique in that it does not specify what you are doing at the place you are visting, but only the fact that you are going somewhere and will return. It is also quite useful if you don't want another person to know what you will be doing (though it's not used that way in this context).
Shouldn't "See you later" be an acceptable answer. From what I understand you should only say this if you're leaving a location and you're actually going to see the person later when you return to that location, like a kid leaving for school in the morning saying "see you later" to there parents.
So, this is one of those "te-forms" I've heard about. How does that affect the verb in play? Does it make it a present continuous verb or something? If "iki masu" is "I go" and "ikitteki masu" is "I am going" it seems to transform what's being said from something you normally do to something you are presently doing. I'd rather have a more involved, thorough, and intelligent understanding of this, however. Could someone clarify the language rules here for me, please?
Te form is used in many different grammar constructions, so the meanings can vary. Present progressive is formed using te-iru form, which is a verb in te-form followed by the verb iru.
行きます - i will go.
行っています - I am going.
行ってきます- i will go and come back
The last one is the te form of iku (to go) followed by the verb kuru (to come) in polite form.
Oddly in English, the so-called present tense of active verbs doesn't really refer specifically to what one is doing right now, but rather to what does in general or on a regular, repeated basis. So, for example, 'I go to school' says nothing about whether at this moment you are either on your way or about to go, etc. For that, we need the present progressive tense (also called continuous), BE + Verb+ing. 'I am going (/leaving) [now / soon ...].'