I understand that these three expressions are an invitation to do something or go somewhere. I would use “давай” as in “come on” or “come with me”. “поехали” and ”пойдём” would be more like “let’s go”, but I think it’s a bit too simplistic.
In which case either of these expressions would be preferred over the other, and why?
What words must we use with each expression (e.g. must be followed by a verb or must be followed by a location)?
Поехали could be used colloquially in the sense of пойдём, regardless of means of movement to be undertaken, like in поехали в кино, а slangier version is погнали...
Давай normally requires a verb, in some cases it could be used with adverbials of destination only, like in давай на футбол, давай в магаз, давай ко мне. This construction along with айда + adverbial of destination can be heard in informal conversations of youth in Soviet era movies, but in my lifetime and in my region it has been vanishingly rare, so it could sound somewhat artificial.
It is though a standard colloquial phrasing for guiding a movement towards certain direction - давай сюда / туда / на|влево/левее / на|вправо/правее / назад / прямо / на север/юг/восток/запад / вниз/ниже/ на|вверх/выше / ближе/дальше
«Давай» is more like "let's", it's not used today without another verb: «давай пойдём» — "let's go", «давай поговорим» — "let's talk" etc. It can be also used as a phrase itself, roughly like "sure": «Поужинаем вместе? — Давай», "Let's have dinner together? — Sure"
«Поехали» is more like "go!" (not necessarily about moving) and is usually used as a phrase: «Нужен мозговой штурм. Поехали!» — "We need to brainstorm this. Go!"
«Пойдём» is always about moving, and it's somewhat like "let's go", like «Пойдём покурим» — "let's go have a smoke", or «Пойдём в кино» — "let's go to the movies".
