После работы я зайду в супермаркет и заодно куплю кефир.

Because после requires the genitive case in Russian.

• работа = nominative form, “work”
• работы = genitive form

So:

• после работы = after work

This is a very common pattern:

• после урока = after the lesson
• после обеда = after lunch
• после встречи = after the meeting

In this sentence, после работы is an idiomatic way to say after work.

Here it most naturally means after work in the general everyday sense, like after finishing your working day.

Russian often leaves this kind of phrase without anything corresponding to the English article:

• После работы я пойду домой. = After work, I’ll go home.

Depending on context, после работы could also mean after the task/job is finished, but in everyday speech it very often just means after work.

Зайду is the 1st person singular future of зайти.

So:

• я зайду = I will stop by / I will go in / I will drop in

In this sentence, зайду в супермаркет means something like:

• I’ll stop by the supermarket
• I’ll go into the supermarket

It suggests a short visit or a quick stop, not just the general idea of going somewhere.

Because зайти and пойти are different verbs.

• пойти = to go, to set off
• зайти = to go in, to stop by, to drop in

Compare:

• После работы я пойду в супермаркет. = After work I’ll go to the supermarket.
• После работы я зайду в супермаркет. = After work I’ll stop by the supermarket.

Зайду sounds more like a quick, purposeful stop on the way or as part of your routine. It fits very well with заодно куплю кефир.

Because Russian uses в + accusative to show motion into a place.

Here the speaker is moving into the supermarket:

• зайду в супермаркет = I’ll go into / stop by the supermarket

Compare:

• в супермаркет = into the supermarket, to the supermarket
• в супермаркете = in the supermarket

So:

• Я зайду в супермаркет. = I’ll go into the supermarket.
• Я в супермаркете. = I’m in the supermarket.

Because супермаркет is a masculine inanimate noun, and in the accusative singular, that type of noun usually looks the same as the nominative.

So:

• nominative: супермаркет
• accusative: супермаркет

That is why you get:

• в супермаркет

But with other nouns, you may see a visible change:

• в школу = into the school
• в магазин = into the shop
• в комнату = into the room

Заодно means something like:

• while I’m at it
• at the same time
• since I’m there anyway
• as well

In this sentence:

• и заодно куплю кефир = and while I’m at it, I’ll buy kefir

It implies that buying kefir is an extra useful thing to do during the same trip.

A few examples:

• Зайду в аптеку и заодно куплю воду. = I’ll stop by the pharmacy and, while I’m at it, buy water.
• Позвоню ему и заодно спрошу про встречу. = I’ll call him and at the same time ask about the meeting.

Куплю is the future tense of the perfective verb купить.

That means it refers to a single completed action:

• куплю = I will buy

This fits the idea perfectly: the speaker plans to buy kefir once.

Compare:

• куплю = I’ll buy it (one complete purchase)
• буду покупать = I’ll be buying / I’ll buy repeatedly or as a process
• покупаю = I am buying / I buy

In this sentence, the natural choice is куплю, because the action is a simple planned result.

Because both actions happen after the moment of speaking:

1. first, the speaker will stop by the supermarket
2. then, the speaker will buy kefir

Russian often uses perfective future for planned, completed actions in a sequence:

• зайду = I’ll stop by
• куплю = I’ll buy

This gives the sentence a clear sense of planned future events.

Russian has no articles, so it does not use separate words for a/an or the.

That means:

• в супермаркет could mean to a supermarket or to the supermarket
• куплю кефир could mean I’ll buy kefir, I’ll buy some kefir, or I’ll buy the kefir

The exact meaning depends on context.

In this sentence, English would most naturally say:

• After work I’ll stop by the supermarket and buy some kefir while I’m at it.

In Russian, a bare noun often works where English would say some.

So:

• куплю кефир = I’ll buy kefir / I’ll buy some kefir

Russian usually does not need a separate word for some unless the speaker wants to be especially explicit or contrastive.

This is very common with food and drink:

• купить хлеб = buy bread
• купить молоко = buy milk
• купить сыр = buy cheese

Yes. Russian word order is fairly flexible.

The sentence:

• После работы я зайду в супермаркет и заодно куплю кефир.

is natural and neutral. But you could also say:

• После работы я зайду в супермаркет и куплю заодно кефир.
• Я после работы зайду в супермаркет и заодно куплю кефир.

These versions are still understandable, but the original sounds especially smooth and natural.

Putting заодно before куплю is very common, because it clearly signals while I’m at it before the second action.