Для салата я взял тёрку и быстро натёр морковь и сыр.

Because the preposition для always requires the genitive case.

• салат = nominative singular
• салата = genitive singular

So для салата means for the salad.

This is a very common pattern in Russian:

• для супа = for the soup
• для друга = for a friend
• для работы = for work

Because тёрку is the accusative singular form of тёрка.

Here, тёрка is the direct object of взял (took), so it has to go into the accusative case.

For most feminine nouns ending in -а, the accusative singular changes to -у:

• тёрка → тёрку
• книга → книгу
• ложка → ложку

So:

• Я взял тёрку = I took a grater

They are also direct objects, but their accusative forms happen to look the same as the nominative.

сыр

сыр is a masculine inanimate noun. For masculine inanimate nouns, the accusative usually equals the nominative:

• nominative: сыр
• accusative: сыр

морковь

морковь is a feminine noun of the third declension (it ends in a soft sign -ь). For this type of inanimate feminine noun, nominative and accusative are also usually the same:

• nominative: морковь
• accusative: морковь

So in this sentence:

• натёр морковь и сыр = grated carrots and cheese

In the Russian past tense, verbs agree with the subject in gender and number.

Since the subject is я, the form tells you the speaker is male:

• я взял = I took (male speaker)
• я натёр = I grated (male speaker)

If the speaker were female, it would be:

• я взяла
• я натёрла

This is something English does not do, so it often stands out to learners.

Взял is the past tense masculine singular form of взять, which means to take.

So:

• взять = to take
• взял = took (male speaker)

Russian commonly uses взять in everyday actions like taking or picking up an object before doing something with it:

• взял нож = took a knife
• взял ручку = took a pen
• взял тёрку = took a grater

Here it marks the first completed action in the sequence.

This is about aspect.

• взять / взял = perfective
• брать / брал = imperfective

In this sentence, the speaker is describing a sequence of completed actions:

1. he took the grater
2. he grated the carrot and cheese

So perfective взял fits naturally.

Very roughly:

• брал can suggest process, repetition, or background context
• взял presents a single completed action

Here, взял is the normal choice.

Натёр is the masculine past tense form of натереть.

• натереть = to grate / rub up
• натёр = grated / rubbed (male speaker)

It can look surprising because the stem changes:

• тереть
• натереть
• натёр

This kind of vowel change is normal in some Russian verbs.

Also, the letter ё is important here:

• натёр is stressed on ё
• if written without dots, you may see натер, but the correct pronunciation is still натёр

In this sentence, натёр means that the grating was completed.

Because the sentence tells a short story made of two finished actions.

• взял = took
• натёр = grated

Russian often uses perfective verbs in narration when one completed action follows another.

So the sentence feels like:

• For the salad, I took a grater and quickly grated the carrot and cheese.

If imperfective forms were used, the meaning would shift toward process, repetition, or background description.

Russian has no articles.

So nouns like:

• тёрка
• морковь
• сыр
• салат

can mean a or the depending on context.

For example:

• взял тёрку could mean took a grater or took the grater
• натёр морковь и сыр could mean grated carrot and cheese or the carrot and the cheese

You understand which one is meant from context, not from an article.

Быстро is an adverb meaning quickly. It modifies натёр.

Its position here is natural, but Russian word order is more flexible than English. The sentence uses a very normal, neutral order:

• я взял тёрку и быстро натёр морковь и сыр

You could move быстро for emphasis, for example:

• я быстро натёр морковь и сыр
• морковь и сыр я натёр быстро

These are possible, but they may sound slightly different in focus or style. The original order is straightforward and idiomatic.

Because и is simply connecting two predicates with the same subject:

• я взял тёрку
• и быстро натёр морковь и сыр

This is like English:

• I took a grater and quickly grated the carrot and cheese.

No comma is needed in normal Russian punctuation when и joins two parts of a sentence with the same subject.

Yes, very often.

Russian often drops subject pronouns when they are clear from the verb form and context. So you could say:

• Для салата взял тёрку и быстро натёр морковь и сыр.

That is possible, especially in conversation.

However, keeping я is also completely natural. It can make the sentence clearer, more explicit, or slightly more personal. In a learner sentence, including я is especially helpful because it makes the structure easy to see.

Тёрка means grater — the kitchen tool used for grating food.

It comes from the same general root as verbs related to rubbing or grating:

• тереть = to rub
• натереть = to grate / rub up
• тёрка = grater

So the sentence is very tightly connected in vocabulary:

• he took a grater
• and grated the carrot and cheese

That kind of word-family connection is common in Russian.