Grammar JLPT Level N4 - Lesson 1

This grammar construction is a mix of the conjunction particle て, and the verb 行(い)く 'to go'. To use this grammar point, you will need to attach て to any verb, and then follow it with いく.
This nuance is usually that '(A) happened and then went', or 'will happen, and then go'. In other words, it will depend on whether いく is in present, or past tense.

• 今日(きょう)はうちで食(た)べていく？
  Do you want to go and eat at my place tonight? (Literally 'will you', but comes across more as an invitation in casual Japanese)
• 今晩(こんばん)は雨(あめ)が降(ふ)ると聞(き)いたから、傘(かさ)を持(も)って行(い)った。
  Because I heard that it was going to rain tonight, I took an umbrella.

ていく can be used when representing time, or when representing a physical place. When representing a physical place, it is more likely that the kanji structure, 行(い)く will be used. However, this is not a set rule, and depends on the writer.
When used in relation to time, the nuance is usually that something has already started/is about to start, and then will continue to progress in the same way.

• 毎日(まいにち)ゴルフの練習(れんしゅう)をしたらだんだんと上手(うま)くなっていく。
  When you practice golf everyday, you will gradually get better.

This is the opposite of てくる, which indicates that something has finally reached a certain state, after progressing in that way for some time.