Сегодня после работы мы идём в магазин, потому что дочке нужны новая кисточка и новый альбом.

Because после requires the genitive case. The noun работа becomes работы in the genitive singular.

So:

• работа = work, job
• после работы = after work

This is a very common pattern:

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

It can mean we are going in the sense of we’re going / we’re headed / we’re going to go. In Russian, the present tense of motion verbs is often used for a planned near-future action, just like English We’re going to the store later.

So Сегодня после работы мы идём в магазин naturally means something like:

• Today after work we’re going to the store

not necessarily we are walking there right this second.

Because идти / идём is used for one specific trip in one direction, while ходить / ходим is used for repeated, habitual, or multi-directional movement.

Here, the sentence is about one planned trip to the store, so идём is the right choice.

Compare:

• Мы идём в магазин = we’re going to the store now / soon
• Мы ходим в магазин каждый день = we go to the store every day

Strictly speaking, идти is the verb for going on foot or for movement in one direction. So yes, the most basic meaning suggests going by walking.

In everyday speech, though, people sometimes use it a bit more loosely when the exact mode of travel is not important. But if you specifically want to say going by vehicle, Russian often uses ехать instead.

So:

• идём в магазин = going to the store, often understood as on foot
• едем в магазин = going to the store by car/bus/etc.

Because в can take different cases depending on the meaning:

• в + accusative = motion into / to
• в + prepositional = location in / at

Here there is movement toward the store, so Russian uses the accusative:

• в магазин = to the store

Compare:

• Мы идём в магазин = we’re going to the store
• Мы в магазине = we are in the store

Дочке is the dative singular of дочка. The dative is used here for the person who needs something.

Russian often expresses someone needs something with this pattern:

• кому? = to whom?
• что нужно? = what is needed?

So:

• дочке нужны... = the daughter needs...

Literally, it is closer to:

• To the daughter, ... are needed

Because in this construction, the things needed are the grammatical subject of нужен / нужна / нужны, not direct objects.

So Russian builds it like this:

• дочке нужна кисточка = the daughter needs a brush
• literally: to the daughter, a brush is necessary

That is why кисточка and альбом stay in the nominative.

This is very different from English, where need takes a direct object.

Because there are two things being needed:

• новая кисточка
• новый альбом

Together they form a plural idea, so the predicate is plural:

• дочке нужны новая кисточка и новый альбом

If there were only one item, you would get singular agreement:

• дочке нужна новая кисточка
• дочке нужен новый альбом

Because adjectives must agree with the noun in gender, number, and case.

Here both nouns are singular nominative, but they have different genders:

• кисточка is feminine → новая кисточка
• альбом is masculine → новый альбом

So the adjective новый changes form to match the noun.

Дочка is a common everyday word meaning daughter. It often sounds a bit warmer, more natural, or more conversational than дочь.

Very roughly:

• дочь = more neutral, dictionary/basic form
• дочка = very common in speech, often affectionate

In this sentence, дочке sounds very natural in ordinary conversation.

Because потому что introduces a subordinate clause meaning because. In Russian, such clauses are normally separated by a comma.

So the sentence has two parts:

• Сегодня после работы мы идём в магазин
• потому что дочке нужны новая кисточка и новый альбом

The comma marks the boundary between the main clause and the reason clause.

Russian word order is fairly flexible. This sentence uses a very natural, neutral order, but other orders are possible if you want different emphasis.

For example:

• Сегодня после работы мы идём в магазин... = neutral, smooth
• Мы сегодня после работы идём в магазин... = a bit more emphasis on we
• Дочке нужны новая кисточка и новый альбом, поэтому мы идём в магазин = starts with the reason instead

Even though the order can change, the cases help show who is doing what.

Russian has no articles, so it does not directly mark a vs the.

That means в магазин can mean:

• to a store
• to the store

The exact meaning comes from context. In this sentence, English would often naturally use the store, but Russian itself does not spell that out with an article.