На рынке мы взяли по три сливы и по одному персику на десерт.

Here по has a distributive meaning: each, apiece.

So the sentence means that each person got:

• three plums
• one peach

Without по, три сливы и один персик would usually sound like a simple quantity, not clearly for each person.

Russian often repeats по when two separate distributive amounts are listed.

So:

• по три сливы и по одному персику

makes it very clear that both quantities are per person.

If you removed the second по, the meaning might still be understood from context, but repeating it is more natural and clearer.

That is a very common learner question.

With distributive по meaning each/apiece, Russian often uses a special pattern with numerals. For small numbers, the usual forms are:

• по одному персику
• по два яблока
• по три сливы
• по четыре груши

So по три сливы is the normal way to say three plums each.

Also, сливы here is the form used after три, just as in три сливы.

Because with по одному, Russian uses the dative singular.

So:

• один персик = one peach
• по одному персику = one peach each

Both words change:

• один → одному
• персик → персику

This is a very standard pattern:

• по одной книге
• по одному билету
• по одному персику

After два, три, четыре, Russian nouns usually appear in a special counting form that is traditionally described as genitive singular.

So:

• одна слива
• три сливы
• пять слив

For learners, the important thing is to memorize the pattern:

• after 2, 3, 4 → a form like сливы
• after 5 and higher → a form like слив

So три сливы is exactly what you would expect.

Russian idiomatically says на рынке for at the market.

This is just one of those place expressions that uses на instead of в. English speakers often want to translate word-for-word, but Russian chooses different prepositions for some locations.

Here:

• на рынке = at the market / in the market area

Because this is a location, рынке is in the prepositional case.

Yes, взяли is the past plural of взять, which literally means took.

But in everyday Russian, взять can also mean something like:

• took
• got
• picked up
• sometimes even bought, depending on context

In a market context, мы взяли... often sounds natural in the sense of we got / we picked up.

If the speaker wanted to stress the act of buying, купили would also be possible.
Взяли focuses more on the completed choice or acquisition.

На десерт is a very common Russian expression meaning for dessert or as dessert.

Russian often uses на + accusative to show purpose or intended use:

• на завтрак = for breakfast
• на ужин = for dinner
• на память = as a keepsake
• на десерт = for dessert

So here it tells you what the fruit was for.

Мы is not strictly necessary, because взяли already shows we.

Russian often drops subject pronouns when the verb makes the subject clear. So in context, На рынке взяли... could work.

But мы can still be used for:

• clarity
• emphasis
• natural storytelling style

As for word order, Russian is flexible. Starting with На рынке sets the scene first: At the market...

So the sentence structure feels like:

• At the market
• we got
• three plums each and one peach each
• for dessert