Learn Hardcore Russian: Тётя учит нас вежливости, когда кто‑то приходит в гости. - My aunt teaches us politeness when someone comes to visit.

Accusative for the person being taught: учить кого? → нас (us)
Dative for the thing being taught would be English-like, but in Russian, this verb instead uses:
Genitive for the subject being taught: учить чему? → вежливости (politeness)

So the pattern is:

учить кого чему – “to teach someone something”
- учить детей математике – to teach the children math
- учить нас вежливости – to teach us politeness

That’s why it’s нас (accusative) and вежливости (genitive).

Could I say Тётя учит нас быть вежливыми instead of учит нас вежливости? What’s the difference?

Yes, both are correct but slightly different in feel:

Тётя учит нас вежливости.
Literally: “Aunt teaches us politeness.”
This sounds more abstract: she is teaching the quality of politeness, polite behavior in general.
Тётя учит нас быть вежливыми.
Literally: “Aunt teaches us to be polite.”
Here you describe an action or way of behaving, with an infinitive phrase быть вежливыми.

They’re very close in meaning.
The original sentence emphasizes the concept / quality (вежливость), the alternative emphasizes behavior.

Why is there a comma before когда?

In Russian, когда introduces a subordinate clause of time, so it usually forms a separate clause that is divided from the main clause by a comma.

Structure here:

Main clause: Тётя учит нас вежливости
Subordinate time clause: когда кто‑то приходит в гости

Rule: you separate the main and subordinate clauses with a comma, just like “Aunt teaches us politeness, when someone comes to visit.”

Why is it кто‑то приходит, not кто‑то приходят?

Even though кто‑то (“someone”) refers to an unknown person, grammatically it is singular and masculine in Russian.

So the verb agrees with it as with a singular subject:

кто‑то приходит – someone comes
кто‑то позвонил – someone called

Using plural приходят would be incorrect with кто‑то.

What is the purpose of the hyphen in кто‑то?

Indefinite pronouns like “someone, something, somewhere” are often formed with a particle ‑то, which is attached with a hyphen:

кто‑то – someone
что‑то – something
где‑то – somewhere

So кто‑то is the standard spelling; the hyphen is required.

Why is it приходит and not придёт or пришёл?

Приходит is:

present tense
imperfective aspect
3rd person singular

Here it expresses a repeated / habitual action:

когда кто‑то приходит в гости
“when someone comes to visit (whenever that happens, as a rule)”

If you said:

когда кто‑то придёт в гости – “when someone (will) come to visit (on some particular future occasion)”
когда кто‑то пришёл в гости – “when someone came to visit (in the past, on that occasion)”

So приходит matches the idea of a general rule or routine: every time someone comes, aunt teaches us politeness.

What does в гости literally mean, and why is it в гости, not в гостях?

В гости is an idiomatic phrase meaning “(to go / come) as a guest; to visit someone’s place.”

приходить в гости – to come visit (to someone’s home)
идти в гости – to go visiting / go to someone’s place

Literally, гости is the plural nominative “guests”, but in в гости it functions as an adverbial phrase (“on a visit”). Russian freezes this form in this expression.

Contrast:

в гости – direction: “to visit” (movement towards someone’s home)
в гостях – location: “as a guest; at someone’s place”
- Мы в гостях у тёти. – We are visiting our aunt / We are at our aunt’s place.

Why is Тётя capitalized here? Is that always the case?

It’s capitalized here simply because it is the first word of the sentence.

As a common noun:

тётя – “aunt” (not a proper name) – normally written with a lowercase т in the middle of a sentence.

It can be capitalized if used as part of a name or a form of address like a title:

Тётя Оля – Aunt Olya

But in most neutral contexts mid-sentence you would write:

Моя тётя учит нас вежливости.

What is the difference between учить and изучать? Could I say Тётя изучает нас вежливости?

No, Тётя изучает нас вежливости is wrong.

Difference:

учить – “to teach” or “to learn (memorize)”
- учить кого чему – teach someone something
- учить слова – learn/memorize words
изучать – “to study, to examine in depth”
- изучать русский язык – to study Russian
- изучать проблему – to study a problem

Here we need “teach us politeness”, so only учить works:

Тётя учит нас вежливости. ✔
Тётя изучает нас вежливости. ✘ (nonsense in Russian)

Could I put когда кто‑то приходит в гости at the beginning of the sentence?

Yes, that’s perfectly correct and natural. You’d just keep the comma:

Когда кто‑то приходит в гости, тётя учит нас вежливости.

Meaning stays the same.
Word order is flexible; putting the time clause first can slightly emphasize the condition (“Whenever someone comes over, that’s when she teaches us politeness”).

Why is вежливость in the singular? Could I say вежливости in plural meaning “polite behaviors”?

Вежливость is an abstract uncountable noun (like “kindness,” “politeness” in English). Russian almost always uses it in the singular when referring to the quality in general.

In your sentence, вежливости is singular genitive (not plural):

nominative: вежливость
genitive singular: вежливости

There is also a plural вежливости, but that would mean “polite remarks / courtesies” in a very specific sense and would not fit here.

So in учит нас вежливости, it is singular genitive: “teaches us (the quality of) politeness.”