Quick answer
Проходи́те!
prakhadíte — the imperative does the whole job — no “after you” needed
The options
Проходи́те!Do go through! · After you!
Пожа́луйста!Please! — with a gesture, enough on its own
Проходи́те, пожа́луйста.Please go ahead.
Вы пе́рвый.You first.
Разреши́те.Excuse me. — squeezing past
The imperative of invitation
Russian politeness often works through imperatives: проходи́те (go through), сади́тесь (sit down), угоща́йтесь (help yourself), бери́те (take some). All of them are warm invitations, not commands, and all take the polite -те.
Разреши́те, the other direction
When *you* need to get past, the word is разреши́те — allow me — or позво́льте in a more formal setting. In a crowded metro car Разреши́те! is what you say, and people step aside.
What you hear back
Спаси́бо!Thank you!
Нет, вы пе́рвый.No, you first.
Да, спаси́бо.Yes, thanks.
Typical mistakes
По́сле вас.
Проходи́те!
The literal translation is not used.
Проходи́ — to a stranger
Проходи́те.
Use the polite -те form.
Check yourself
How do you say “after you” at a door?
The invitation imperative.