Quick answer
Дава́й пойдём!
davay poydyóm — and simply Дава́й! — the word this site is named after
The options
Дава́й!Come on! / Let’s! — the all-purpose one
Пошли́!Let’s go! — casual, very common
Пойдём!Let’s go. — neutral
Идём!Let’s go — we’re leaving now
Пое́хали!Let’s go! — by transport, and Gagarin’s word in 1961
Пое́хали!
When Gagarin left the ground in 1961 he said Пое́хали! — “off we go”. It is still what Russians say when anything starts moving, from a car to a project.
Дава́й as a goodbye
On the phone, Ну, дава́й! simply means “right, bye then”. The verb дава́ть has drifted a long way from “to give”.
What you hear back
Пошли́!Let’s go!
Дава́й.All right.
Мину́тку!Just a minute!
Typical mistakes
Дава́йте идти́!
Дава́йте пойдём! / Пошли́!
дава́й works with the perfective мы-form, not with a plain infinitive of motion.
Пошли́ to your boss
Пойдёмте!
The -те form keeps it polite.
Check yourself
What did Gagarin say at launch?
Пое́хали! — and Russians still use it for any start.