Can someone explain the differences between v-ていく and v-てくる for me. I know that they both express some kind of ongoing action (like a place getting crowded). For example, what's the difference between 込んでいく and 込んでくる, or is it even possible to use both of these. Also, do you generally use kanji to write these?
Most of the examples I can think of use v-てくる, like 雨がやんできた, or 空が明るくなってきた. How does the meaning change if I use いく instead?
Edit: changed て行く／て来る to the more proper ていく／てくる.
～ていく and ～てくる (usually written in kana, since they are such common suffixes) can express both physical movement (such as in 行【い】 ってくる "go and come back") or a continued change in state. Since your question regards the latter usage, I'll restrict my answer to that.
To use your examples:
雨【あめ】がやんできた。 The rain [over a period of time up until now] stopped.
雨【あめ】がやんでいった。 The rain [over a period of time up until some time before now] stopped.
With the past tense, the point in time at which the action ends, relative to the present moment, changes depending on whether きた or いった is used.
空【そら】が明【あか】るくなってくる。 The sky [over a period of time through now and continuing to some future point] brightens.
空【そら】が明【あか】るくなっていく。 The sky [over a period of time starting now and continuing indefinitely] brightens.
With the present tense, the factor that changes is the point in time at which the action begins. (～てくる typically implies the action started in the past, while ～ていく typically implies the action starts now.)
Any verb that indicates a change in state is a prime candidate for the ～ていく and ～てくる suffixes. なる, 増【ふ】える, 減【へ】る, and 広【ひろ】がる are a few examples.
As a postscript, I should also mention that the choice between these two can depend also on whether "now" means actual now versus a reference point in the past or future. The diagram on page 120 of this text may help clear this up.
Both means a continued change in state but with slight difference.
行く is for something that moves away from the speaker (not necessarily a physical movement), or away from another person's viewpoint that the speaker adopts.
来る is for something that moves towards the speaker (not necessarily a physical movement), or towards another person's viewpoint that the speaker adopts.
Apart from temporal differences already explained by Derek, it also can show the speaker's involvement.
The 来る versions imply that the change is somehow going to involve the speaker or has already involved the speaker. (towards speaker)
The 行く versions feel impersonal and objective (away from speaker)
〜ていく means "will go/get". So it means from the current time onward. 〜てくる mean "came/got to be", as in from some time in the past up until now.
→ 〜てくる or 〜ていく →
Note that with these two patterns, you usually write いく／くる in hiragana.
