When we talk about greetings, we often want to talk about the first time you meet someone.

When Americans meet each other for the first time, they often shake hands.

This sentence is only talking about the first meeting. It does not say anything about the second time two people meet. It is not correct to say:

When Americans meet at first, they often shake hands. (WRONG!)

This sentence means, “Every time Americans meet, the first thing they do is shake hands.” At first means at the beginning of a time.

When I meet new people, at first I am very shy, but later I become more relaxed. (CORRECT)

It was raining at first, but then it stopped. (CORRECT)

It is also correct to say:

When Americans first meet, they often shake hands. (CORRECT)

But be careful, it is NOT correct to say “meet first”:

When Americans meet first, they shake hands (WRONG!)

If first is just before the verb it is used to talk about the first time something happen.

I first played the piano when I was a child. (the first time)

When I first saw her new haircut, I was very surprised. (the first time)

If first is NOT just before the verb, it is usually talking about sequence or position.

I played the piano first, then I played the drums. (sequence of events)

My first daughter was born in May; my second daughter was born in July. (position)

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