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давай can mean many things, the most common ones are "Let's" and "Come on"

Давай пойдем в кино! - Let's go to the cinema (notice verb of motion needed for "let's go")
давай, быстрее! - Come on!...

I think Пойдём! is not used as often as Пошли!.
They say this movie is very good. Давай пойдём завтра!
Hurry, were are late. Пойдём, пойдём, enough of sightseeing!

You can treat it like two different words.
Пошли! - Let's go!
Вечером они пошли в кино. - In the evening they went to the movies.