来る is used as a helping verb, aside from its main meaning of “to come”. The below usages are all extremely common and can be the difference between your Japanese sounds natural or not.
来る is used to express one’s intention for the immediate future, when physical movements of going somewhere, do something and come back are involved.
- ちょっとトイレに行って来ます -> I’m going to bathroom real quick
- ちょっとトイレに行きます -> Not wrong, but native speakers never say this
- ちょっと散歩して来る -> I’m going to take a walk outside
- 食べ物を買って来る -> I’m going to do grocery shopping
彼に電話して来る-> Doesn’t involve physical movements (unless the only available phone is located away)
来る is also used for actions that have been continuing to that point.
- 彼はこの会社で四十年間働いて来た -> He has been working for this company for 40 years
- この大学は沢山のノーベル賞学者を輩出して来た -> This university has been producing many Nobel prize winners
し続けてきた (=する＋続ける＋来る) sounds redundant, but very commonly used.
- 努力して来た甲斐があった
- 努力し続けて来た甲斐があった -> Basically the same with the above, maybe a little more emphasis
- I’ve been working hard and I got rewarded
By the way, as you may have already guessed, actions that have been continuing from the past and will continue into the future is expressed with 行く.
- これからも一生懸命頑張って行く -> I will keep working hard
Another usage of 来る is “to gradually become”.
- 十月になって寒くなって来た -> It is October and it has gotten colder
- 林檎が熟して来た -> The apples are getting ripe
- だいぶ日本語が話せるようになって来た -> I’m getting a lot better with speaking Japanese
- 相対性理論が解って来た -> I’ve started understanding the relativity theory
- 彼の文句には本当にうんざりして来た -> I am getting really fed up with his complaints
- 精神的に疲れて来た -> I am getting mentally worn out
As you can see, many of what you express with “be doing” and “have done” in English are expressed using 来る in Japanese.