Мой брат взял лыжи, а сестра сказала, что ей больше нравятся коньки.

Russian often leaves out possessive words like мой / моя / моё / мои when the meaning is already clear from context.

So:

• Мой брат = my brother
• сестра here still means my sister because the sentence is clearly talking about the same family

You could also say а моя сестра сказала..., but it is not necessary.

Because мой has to agree with the gender and number of the noun:

• брат is masculine singular
• so the correct form is мой

Compare:

• мой брат
• моя сестра
• моё окно
• мои друзья

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

In Russian past tense, the verb agrees with the subject in gender and number:

• брат взял = masculine singular
• сестра взяла = feminine singular
• они взяли = plural

So брат взял means the brother took.

Because the subject is сестра, which is feminine.

In the past tense:

• masculine: сказал
• feminine: сказала
• neuter: сказало
• plural: сказали

So:

• сестра сказала = the sister said

А often links two parts of a sentence by contrast or comparison.

Here the idea is something like:

• My brother took skis, whereas my sister said...

So а works well because it sets the brother and sister against each other in a mild contrast.

Compare roughly:

• и = and
• но = but
• а = and / while / whereas, often with contrast

In this sentence, а is the most natural choice.

Лыжи means skis. Here it is the direct object of взял.

For inanimate plural nouns, the accusative case is often the same as the nominative case. So:

• nominative: лыжи
• accusative: лыжи

That is why the form does not change here.

A single ski is лыжа, but in normal use people usually talk about skis as a pair, so лыжи is much more natural.

Similarly, коньки means skates, usually understood as a pair.

So in this sentence:

• взял лыжи = took the skis
• нравятся коньки = likes skates

Что here means that and introduces a subordinate clause.

So:

• сестра сказала, что... = the sister said that...

In English, that is often optional:

• She said that she likes skates more
• She said she likes skates more

In Russian, что is normally used in this kind of sentence.

Because the verb нравиться works differently from English to like.

Russian expresses this idea more like:

• Skates are pleasing to her

So the person who experiences the liking goes in the dative case:

• мне нравится = I like
• тебе нравится = you like
• ей нравятся = she likes

That is why we use ей (to her) instead of она (she).

Because the grammatical subject of нравиться is коньки, and коньки is plural.

With нравиться, the thing that is liked controls the verb:

• ей нравится книга = she likes the book
• ей нравятся книги = she likes the books

Here:

• коньки = plural
• so the verb must be нравятся

Больше means more, so ей больше нравятся коньки literally means:

• skates please her more

In natural English, this usually becomes:

• she likes skates better
• she prefers skates

So больше нравиться is a common way to express preference.

Russian word order is more flexible than English word order. The sentence could be rearranged in different ways, but each version gives a slightly different emphasis.

Ей больше нравятся коньки is a very natural order. It places коньки at the end, which can make them the important new information.

You might also hear:

• Коньки ей больше нравятся
• Ей коньки больше нравятся

These are all possible, but the original version is neutral and natural.

It most directly means took. Depending on context, English might translate it in different ways:

• took
• picked up
• grabbed

In a situation where someone is choosing sports equipment, взял лыжи could also sound like went with skis or chose skis, but the literal meaning is still took skis.

Yes, that is the normal pattern.

Russian нравиться works like this:

• Мне нравится фильм
  Literally: The film is pleasing to me
• Мне нравятся фильмы
  Literally: Films are pleasing to me

So the structure is:

• person in dative
• thing liked in nominative
• verb agrees with the thing liked

That is exactly what happens in:

• ей больше нравятся коньки

You can do a very literal breakdown, but it will sound unnatural in English:

• Мой брат = my brother
• взял лыжи = took skis
• а сестра сказала = and/whereas the sister said
• что ей больше нравятся коньки = that to her skates are pleasing more

A natural English translation would be something like:

• My brother took the skis, and my sister said she liked skates better.
• My brother chose skis, while my sister said she preferred skates.

So the grammar does not match English word for word, especially in the ей больше нравятся коньки part.