После прогулки мы взяли по одному пончику и пошли домой.

Because после requires the genitive case.

The dictionary form is прогулка. Its genitive singular is прогулки.

So:

• прогулка = a walk, stroll
• после прогулки = after a walk / after the walk

This is a very common pattern in Russian:

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

It can mean either, depending on context.

Russian has no articles like a and the, so прогулки by itself does not tell you whether the walk is definite or indefinite. The context decides that.

So После прогулки can be understood as:

• after the walk
• after a walk
• after going for a walk

In this sentence, English would often naturally say After the walk or After our walk.

Because взяли comes from взять, which is perfective. It presents the action as a completed whole: they took/got the doughnuts and that action is finished.

Compare:

• взяли = took, got, picked up
• брали = were taking, used to take, took repeatedly, or took with focus on the process/context rather than completion

In this sentence, the speaker is telling a sequence of completed events:

1. after the walk,
2. we took one doughnut each,
3. and went home.

So the perfective взяли fits very naturally.

Because they are past tense plural forms.

In Russian past tense, verbs agree with the subject in gender/number, not in person the way English does.

Since the subject is мы = we, the verbs must be plural:

• взял = he took
• взяла = she took
• взяло = it took
• взяли = they/we took

And similarly:

• пошёл = he went
• пошла = she went
• пошло = it went
• пошли = they/we went

So мы взяли ... и пошли ... = we took ... and went ...

It means one doughnut each or one doughnut apiece.

The key word here is по, which often has a distributive meaning: something is distributed among people one by one, two each, etc.

So:

• мы взяли по одному пончику = we each took one doughnut

This is different from:

• мы взяли один пончик = we took one doughnut

That second version would usually mean one doughnut total, not one for each person.

Because after по in this kind of distributive expression, Russian uses special case forms.

Here:

• одному = dative singular of один
• пончику = dative singular of пончик

So the pattern is:

• по одному пончику = one doughnut each

This is a very common construction:

• по одному яблоку = one apple each
• по одному билету = one ticket each
• по одному вопросу = one question each

For a learner, the most important thing is to recognize the whole chunk по одному + noun as meaning one ... each.

It can suggest both, but in this sentence the natural translation is simply went home.

Пошли is the past plural of пойти, a perfective verb. It often emphasizes the beginning of movement: set off, started going. But in many everyday sentences, English just translates it as went.

So:

• пошли домой = went home / set off for home

Russian often uses пойти in narratives to mark the next completed action in a sequence.

If you used шли домой, that would focus more on the process of being on the way home:

• мы шли домой = we were walking home / we were on our way home

But пошли домой moves the story forward: they took the doughnuts and then headed home.

Because домой is the normal Russian word for homeward / home when talking about movement toward home.

So:

• идти домой = to go home
• поехать домой = to go home
• вернуться домой = to return home

Compare:

• домой = home, toward home
• в дом = into a house/building
• к дому = toward the house/building

So пошли домой means they went to their home, not merely toward some building.

Yes. Russian word order is fairly flexible.

The sentence as given is neutral and natural:

• После прогулки мы взяли по одному пончику и пошли домой.

But you could also say:

• Мы после прогулки взяли по одному пончику и пошли домой.

This still means the same thing. The difference is mostly about emphasis and flow.

Russian often puts time expressions like После прогулки near the beginning to set the scene first: After the walk...

Because simple и is enough.

Russian often tells a sequence of actions just by linking perfective past verbs with и:

• взяли ... и пошли ... = took ... and went ...

The order of the verbs already shows the order of events, so no extra word is necessary. In English, we might sometimes say and then, but Russian often leaves that implied.