I often met 3 adverbs of time in English : "at first", "initially" and "at the first time/ the first time", but i can't make a distinction. They are the same and can be used interchangeably in any situation, right? If there are any difference between them, please give a some examples.

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    Get used to the fact that English has many different ways of saying the same thing. The three expressions you list all mean the same thing. One caveat: I've never, ever heard anyone say "at the first time"; it's always just "the first time" ... but you could say "at the first moment" and the like. For time expressions it's usually in or on: "in the first minute" or "on the first day," etc. – Robusto Nov 24 '16 at 1:24

My understanding of your time phrases is

at first = in the beginning
a small period of time after something started or the start

initially = at the start
a less small period of time after something started

for the first time / the first time
refers to the maiden occurrence of something

Diagram of time phrases (not drawn to scale)

beginning                                              end
|--------------- duration of an event ------------|
|- at first -|
|----- initially -----|
|-----------------for the first time -----------------|


"At first" and "initially" are interchangeable. They're both adverbs meaning "at the start of something". "Initially" is more formal.

Like most adverbs, "initially" can appear almost anywhere in a sentence. "At first" can too, but in practice, it usually comes at the beginning or end of a clause.

Note that the adjectives "first" and "initial" are equally interchangeable.

"The first time" has both a subtly different meaning and a different grammatical function. It refers to the entire duration of the first occurrence of a repeated event.

"The first time" can be used in a limited number of roles as an adverb, but more commonly it plays a special role as a subordinator: It comes at the beginning of a clause and forces it to modify another clause.

The first time I met him, he was playing basketball.

"I met him" could stand as a sentence on its own. But "The first time I met him" can't; it's subordinate to "he was playing basketball."

The clauses can also be switched without changing the meaning:

He was playing basketball the first time I met him.

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