行く (to go) and 来る (to come) with て creates constructions that are not as straightforward as one would expect. What dictates the usage of ～ていく and ～てくる is highly context driven. It's rather impossible for even the best linguists to properly define all of the possible situations these phrases can be used in. "Highly context driven" refers to the fact that how you interpret them is not just determined by the transitivity of the verb phrase but by what exactly it is "semantically".
Categorical properties of what exactly the verb means is very important to keep in mind, and this also means that you can't just assume that English and Japanese match perfectly on a word-by-word basis. Usually, even this is never the case. After all, the two languages are NOT related to each other.
When physical action is involved, ～ていく and ～てくる show action away and action towards the speaker’s current location respectively. In this case, the verbs are typically written in 漢字, but this is not always the case. At times, these phrases result in sentences that do not sound quite like anything one would say in English. In such cases, think really hard about the real world context in which they are used to better understand the semantic properties at work.
I'm going out for a bit (and will come back).
I went to the park by foot.
Grammar Note: Remember that some て phrases act like adverbs in Japanese, which then can then like in this sentence equate to a prepositional phrase, in this case 歩いて = by foot.
I went having eaten breakfast.
I'm going and coming back.
Usage Note: The above phrase is used when leaving your residence, knowing that you will be coming back. Though the plain form is acceptable with those you are close to, given its status as a set phrase, the polite form is also used without any repercussions.
I'm going to get a chair from the kitchen.
If I have but 10 minutes, I'll come home.
When there is no physical motion involved, ～ていく indicates disappearance whereas ～てくる indicates a process of emergence. Both can show a process of change or continuation, but they are different in this manner. ～ていく would indicate something that will change and continue on into the future whereas ～てくる would represent that has been happening for something and extends to the present.
Mr. Mills has been getting better in Japanese, hasn't he?
Your English will surely get better!
Your desire to want to selfishly choose will surely disappear.
The prisoners came filtering out of the cave.
～てくる can also indicate the inception of a process. As this section will show, understanding this can get really tricky. So, pay attention to detail and be open to differences between Japanese and English grammar.
It started to snow.
Tense can be problematic in making ～てくる and ～ていく seem interchangeable. The semantic differences between the verbs of these phrases and differences in the main verbs used are important. For instance, the fact that 消える is used below is very important.
The fire in the fireplace is about to go out.
The first option indicates inception of a change. The status of the fire gets changed, and the process is "going out". With ～た, you indicate that the process has just started. A strong wind, cutting off of fuel source, or lots of water put on it would determine how quickly it goes out. Nevertheless, the English can be used in the same situations. The second option indicates a process of disappearance. The flame will shortly start to fade and then go out. In a real world situation, it would be very hard to tell all these details, especially because these judgments are all subjective to what the speaker feels.
When considering the English equivalent, you should be able to see how it could be used for both. For brevity, more literal/less natural English equivalents are not shown. With the information given thus far, you should be able to construct more literal interpretations at your own will.
Consider a less puzzling example. Here we have a sentence in reference to the weather. This,again, is an important detail to keep in mind.
It will become gradually hotter.
In this case, you could say that the former indicates an inception of a process that is to happen. It's more vivid and doesn't feel that it's going to take long for this change to occur. Once it has become quite hotter, it could just reach a point and stay that way. The latter option doesn't imply this. Rather, the latter shows a slower, gradual change that will occur and continue on that way. So, even if both sentences were used in the context of the temperature reaching 100F, with the latter, it could just continue to steadily get higher.
Variant Note: It is also important to realize that ～ていく is often contracted to ～てく in casual speech in the same way ～ている is contracted to ～てる. Be sure to recognize it in different conjugations. For example, ～てった ＝ ～ていった. However, this contraction is avoided in conjugations in which it might get confused with ～てくる. For instance, ～ていきなさい would accidentally become ～てきなさい if you dropped い.
I have brought up my children alone.
Just wait here until I come back.
To wait for the teacher to come in.
The ice steadily approached and came (here).
He forgot about bringing the newspaper.
It looks like it’s becoming colder.
Word Note: 回る in this context emphasizes the circulation of gas.
Approaching from behind the curtain like that is scary.
Time started to pass by like an arrow.
He walked to the station through the rain.
Please bring it.
Please take it (away).
I didn't bring a single pen.
I took a silver dollar as a gift.
I forgot to bring some money. Could you lend me some?
Just as I thought it had started to snow, it had already stopped.
Phrase Note: かと思うと ＝ "just as I thought".
Please bring him．
Take him away.
I brought Mr. Okada along tonight.
If you're going to the cinema complex, can't you take me along too?
Word Note: Notice the contrast in definitions between 連れて[いく・くる] and 持って[いく・くる]
I came back home after five days after he left Tokyo.
Normal daily life problems will pile on.
After I planted the sunflower seeds, they sprouted in no less than a week.
She moved to Kobe last month.
I'd like to continue to do my best from now on.
41b. He rushed into my room.
～ていく → ゆく
～ゆく may seldom follow the stem of verbs in a poetic fashion. This is in fact the original form the pattern took, which is why it is deemed poetic/nostalgic. ～てゆく also exists, which is quite nostalgic and is very common in songs and literature.
Even important memories will slip from your memory.
Flowers, burn away.
To be dying.
なってくる VS なっていく
Although the previous information may be sufficient enough to make the differences between these two phrases easy enough to ascertain, given that they are still often misunderstood, it's best to go through this as an independent point of discussion. First, consider the following.
Since it has become cold like this here, it will definitely get steadily cold there after this.
This is a wonderful example of how the two can even be seen in the same complex sentence but yet have zero interchangeability. The first part states the change that it has become cold and still is cold at the speaker's location. In saying so, the speaker makes a conjecture that it will get cold somewhere else away from the speaker. These distinctions are crucial to keep in mind. Here are some more sentences to consider.
49a. 10月だ。故郷は、もう寒くなってきただろうか。 〇
It's October. Hasn't it already gotten cold at our hometown?
Since it's already the middle of May, it should be getting hot soon, right? Once the rainy season lifts, that alone should make it get hotter gradually.
Although I've been on this diet for 6 months, I've only continued to gain and haven't gotten a bit skinnier!
Though the last may be harder to follow from the free translation, it would be a great mistake to change the speech modals to one or the other. When used with intransitive verbs that express change, you are showing a natural change of events by a given process. When you use ～ていく, you say that at the time the change starts, there will continue to be change from that point onward. ～てくる gives a more punctual feel to the instant of change. So, ～ていった would be like "had been getting..." whereas ～てきた would be like "got".