After I have learned English for three months, I met an American on a street.
After learning English for three months, I met an American on a street.
Which one is right in grammar, or more accurate?
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I'm not sure, but it might be worth pointing out, that in English, we don't typically use "learned" as a synonym for "studied." So even after correcting your grammar, your meaning might be getting lost.
Anyway, I think what you might be trying to say is this: "After I'd been studying English for three months, I met an American on the street."
When we use the verb "learn" in the past tense, we usually mean something we have mostly mastered. It is usually only while we are still in the process of learning that we use it as a synonym for studying.
In your sentence you would have to use "after learning", because you use the past tense for "met".
"After" + present perfect indicates a verb that will happen in the future:
After I have learned English, I will visit the UK.
After this pot of spaghetti has boiled for seven minutes, it must be drained.
"After" + gerund can go with any logical time. It doesn't insist on when the thing happened, happens, or will happen; all we know is that the next verb takes place after that.
After learning English, I met an American on the street.
After eating breakfast, I usually go for a run.
After buying a plot of land, I will build a cottage.
In fact, it works like a noun phrase ("after my English course", "after breakfast", "after the purchase").
This also works with a present progressive perfect:
After having learned English, I met an American on the street.
For your sentence, you could also use "After" + simple past, or "After" + past perfect. This indicates a verb that happened in the past.
After I learned English, I met an American on the street.
After I had learned English, I met an American on the street.
There are other threads dealing with the differences between these two tenses, but for now I'll just say that I find the second one a little more natural for this context.