После первой части начался антракт, и мы пошли в фойе за водой.

Because после requires the genitive case.

• первая часть = the first part in the nominative
• After после it changes to первой части

So:

• после концерта = after the concert
• после фильма = after the film
• после первой части = after the first part

Both words change because the adjective must agree with the noun:

• первая → первой
• часть → части

Часть is a feminine noun, and here it is in the genitive singular because of после.

Its basic form is:

• часть = part

Genitive singular:

• части

This is a very common pattern for feminine nouns ending in -ь:

• ночь → ночи
• дверь → двери
• часть → части

So после первой части literally means after the first part.

This is about aspect.

• начаться = to begin, to start up once, as a completed event → perfective
• начинаться = to be beginning / to begin repeatedly or as a process → imperfective

In this sentence, the speaker is talking about a single event that happened:

• антракт начался = the intermission began

So the perfective verb начаться is the natural choice.

Compare:

• Спектакль начался в семь. = The performance started at seven.
• Спектакль начинался, когда мы вошли. = The performance was beginning when we entered.

Both are possible. Russian word order is flexible.

• Начался антракт puts a little more focus on the event beginning.
• Антракт начался is more neutral if you are simply stating the subject first.

In this sentence, После первой части начался антракт sounds very natural and smooth in narration: first the time frame, then what happened.

So this is mainly a matter of information flow, not a grammar difference.

Антракт means intermission, interval, or entr'acte, especially in a theater, opera, or concert setting.

It is the break between parts of a performance.

So in this sentence:

• После первой части начался антракт = After the first part, the intermission began.

This word is common in formal performance contexts. In more everyday situations, a speaker might also say перерыв for a break, but антракт is specifically the kind of break you get during a performance.

Again, this is aspect and meaning.

• идти / шли = to go, to be going, movement as a process
• пойти / пошли = to set off, to start going

Here the idea is that we went off / headed to the foyer after the intermission started. That is a single completed action, so пошли is natural.

Compare:

• Мы пошли в фойе. = We went / headed to the foyer.
• Мы шли в фойе, когда встретили друзей. = We were walking to the foyer when we met friends.

So пошли focuses on the fact that they set out.

After в meaning movement into/to a place, Russian normally uses the accusative case.

So logically this is:

• пойти в + accusative

However, фойе is one of those nouns that is usually indeclinable, meaning its form does not change.

So:

• nominative: фойе
• accusative: фойе
• prepositional: в фойе
• and so on

That is why it looks unchanged.

За has several meanings depending on context and case.

Here, with a verb of motion, идти / пойти за чем-то often means:

• to go get
• to go fetch
• to go for

So:

• пойти за водой = to go get water / to go for water

This is a very common pattern:

• сходить за хлебом = go get bread
• пойти за врачом = go fetch the doctor
• зайти за другом = stop by to pick up a friend

So in this sentence, за водой does not mean behind water. It means to get water.

Because in this meaning, за takes the instrumental case.

Base form:

• вода = water

Instrumental singular:

• водой

So:

• за водой = for water / to get water

This is a fixed and very common case pattern with за in the sense of fetching or going after something.

Compare:

• Я пошёл за кофе.
• Она вышла за билетами.
• Мы пошли за водой.

No, they are different.

• за водой = to get water, to fetch water
• для воды = for water, in the sense of intended for water

Examples:

• Мы пошли за водой. = We went to get water.
• бутылка для воды = a bottle for water

So in your sentence, за водой expresses purpose of movement: they went somewhere in order to get water.

Because this sentence has two clauses, each with its own verb:

1. начался антракт
2. мы пошли в фойе за водой

They are connected by и, so Russian punctuation normally uses a comma here:

• После первой части начался антракт, и мы пошли в фойе за водой.

This is similar to English punctuation in many cases, although Russian tends to use commas quite systematically between clauses.

Yes, you could, but the meaning and feel are slightly different.

• начался антракт = the intermission began
• был антракт = there was an intermission / an intermission took place

The original sentence focuses on the start of the intermission, which fits well with the next action:

• the intermission started,
• and then we went to the foyer

So начался антракт is a bit more dynamic in this context.

Usually, yes, in this kind of context it means the first part/section of a performance.

Depending on the event, it could refer to:

• the first half of a concert
• the first act or section
• the first part of a program

Russian often uses часть quite naturally in these contexts. If the structure were specifically theatrical, you might also see words like:

• акт = act
• отделение = section, part of a concert program

But первая часть is perfectly normal and clear.

Yes, because фойе is indeclinable, the form looks the same even when the meaning changes.

Normally Russian distinguishes:

• в + accusative = into/to
• в + prepositional = in

With фойе, both forms look identical:

• Мы пошли в фойе. = We went into/to the foyer.
• Мы были в фойе. = We were in the foyer.

So you understand the meaning from the verb and the context:

• пошли clearly shows movement toward the foyer
• были would show location in the foyer