После тренировки мы взяли по бутылке воды.

Because после always takes the genitive.

• после = after
• dictionary form: тренировка = training / workout
• genitive singular: тренировки

So:

• после тренировки = after the workout / after training

This is a fixed grammar pattern in Russian:
после + genitive

Examples:

• после урока = after the lesson
• после работы = after work
• после фильма = after the film

Взяли is the past tense plural form of взять, which is a perfective verb meaning to take.

It is plural because the subject is мы = we.

So:

• я взял / взяла = I took
• ты взял / взяла = you took
• мы взяли = we took

In this sentence, взяли can sound natural in English as:

• we took
• we got
• we grabbed

depending on context.

Because по бутылке means one bottle each.

This is a very common Russian pattern:

• по + dative singular often expresses distribution:
  • one each
  • apiece
  • per person

So:

• мы взяли бутылку воды = we took a bottle of water
  (possibly just one bottle total)
• мы взяли по бутылке воды = we each took a bottle of water

That small word по changes the meaning a lot.

Other examples:

• дали по яблоку = they gave an apple each
• купили по билету = they bought a ticket each
• выпили по чашке чая = they drank a cup of tea each

Because after по in this distributive meaning (each / apiece), Russian usually uses the dative singular.

The noun is:

• dictionary form: бутылка
• dative singular: бутылке

So:

• по бутылке = a bottle each

This is not random; it is part of the grammar pattern:

• по книге = one book each
• по чашке = one cup each
• по машине = one car each

Because бутылка воды is a standard container/quantity expression, and in Russian the thing inside the container is usually put in the genitive.

So:

• бутылка = bottle
• воды = of water

Literally, this is a bottle of water.

This is the same pattern as:

• чашка чая = a cup of tea
• стакан молока = a glass of milk
• кусок хлеба = a piece of bread
• бутылка вина = a bottle of wine

If you said воду, that would be accusative and would not fit this structure.

Yes, usually it means one bottle each.

That is the normal interpretation of по + dative singular in sentences like this.

So the sentence means that each person in the group took one bottle of water.

If the speaker wanted to emphasize a different quantity, Russian would normally say it more explicitly, for example:

• по две бутылки воды = two bottles each
• по три яблока = three apples each

Russian has no articles like English a / an / the.

So nouns appear without articles, and context tells you whether the meaning is:

• a bottle
• the bottle
• bottles

In this sentence:

• по бутылке воды is understood as a bottle of water each

Russian relies on context, word order, and sometimes intonation instead of articles.

Yes. Russian word order is more flexible than English word order because case endings show the grammatical relationships.

The neutral order here is:

• После тренировки мы взяли по бутылке воды.

But you could also hear things like:

• Мы после тренировки взяли по бутылке воды.
• По бутылке воды мы взяли после тренировки.

These alternatives may sound more marked or emphasize different parts of the sentence, but the basic meaning stays the same.

For a learner, the original order is a very natural and standard one.

Because взять is perfective, and this sentence describes a completed action: after the workout, they took/got a bottle each.

Russian often uses:

• брать = imperfective, process / repeated action / general action
• взять = perfective, single completed action

So here:

• взяли = they completed the action of taking the bottles

If you used брали, it would sound more like:

• they were taking
• they used to take
• they took repeatedly
• or the action is being viewed as a process rather than a completed event

It can be either, depending on context.

• после тренировки literally means after training / after the workout
• In many real situations, English would naturally translate it as:
  • after practice
  • after the workout
  • after training

Because the meaning is already known from context, the important grammar point is that тренировки is genitive after после.

Yes, very naturally.

Мы взяли по бутылке воды normally suggests that we each took a bottle for ourselves.

Russian does not need to add anything extra to make that the default interpretation here. If the sentence meant they took bottles for other people, the context would usually make that clear.

So the most natural reading is:

• after the workout, we each got ourselves a bottle of water