The neutral I and you: tôi, bạn, and when they sound cold
I / you (neutral)
Pattern: tôi · bạn · các bạn
In Saigon: tôi; tui · In Hà Nội: tôi
Typed without accents: toi, ban
Tôi is the neutral "I": it says nothing about age, and it keeps a polite distance. That makes it right for formal settings, at an office counter, with a stranger whose age you can't place, in speeches and in writing. Bạn (literally "friend") is the neutral "you": for people your own age, in classrooms, and in writing for the public, such as ads and websites. Các bạn is "you all".
Between people who know each other, though, both can sound cold. With someone older, tôi can sound stiff or even rude, as if you're refusing the relationship; say em or con instead. Bạn to someone clearly older sounds odd too: use anh, chị, cô or chú.
Saigon speakers also say tui, and write it in messages. It's casual and friendly, for people you're at ease with: Tui cũng vậy. Me too. In Hà Nội it stays tôi.
When you don't know the person's age and the setting is polite, tôi is safe. Once you know who's older, switch to the family words.
Examples
Highlighted words are what Saigon and Hà Nội say differently.
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Xin lỗi, tôi không hiểu.
Sorry, I don't understand.
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Tôi muốn mở tài khoản ngân hàng.
I'd like to open a bank account.
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Bạn tên là gì?
What's your name?
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Saigon
Tui cũng thích ăn cay.
Hà Nội
Tôi cũng thích ăn cay.
I like spicy food too.
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Chào các bạn!
Hi, everyone!