watasi no ane ha ryouri ga zyouzu desu.

の links two nouns and often shows possession or relationship.

So:

• 私 = I / me
• 姉 = older sister
• 私の姉 = my older sister

In English, we use my. In Japanese, you often use noun + の + noun instead.

姉 is the plain word for my older sister when talking about her to someone else.

Japanese often uses different words depending on whether you are:

• talking about your own family, or
• addressing / referring politely to someone else’s family member

A common contrast is:

• 姉 = my older sister
• お姉さん = someone else’s older sister, or a polite/friendly way to address an older girl/woman

So in this sentence, 私の姉 is natural because the speaker is talking about their own sister.

は marks 私の姉 as the topic of the sentence.

So the sentence is structured like:

• As for my older sister, cooking is something she is good at.

This does not mean は simply equals is. It tells the listener what the sentence is about.

So:

• 私の姉は = as for my older sister

This is a very common Japanese pattern:

• A は B です
  where A is the topic, and then the sentence says something about it.

This is one of the most common learner questions.

In 料理が上手です, が marks the thing in relation to the adjective-like word 上手.

• 料理 = cooking / cuisine
• 上手 = skilled, good at
• 料理が上手 = good at cooking

With words like 上手, 下手, 好き, 嫌い, and sometimes 得意, Japanese often marks the thing involved with が.

So:

• 私の姉は = as for my sister,
• 料理が上手です = cooking is something she is good at

In more natural English, we translate it as:

• My older sister is good at cooking.

Because in this sentence, 料理 is not the main topic. 私の姉 is the topic.

Compare:

• 私の姉は料理が上手です。
  = As for my sister, she is good at cooking.

If you said:

• 料理は上手です。

that would sound more like:

• As for cooking, (someone) is good at it or
• Cooking is good

which is not the intended structure here.

So は sets the overall topic (my sister), while が marks what she is good at (cooking).

It behaves a lot like an adjective in meaning, but grammatically it is usually treated as a na-adjective (or an adjectival noun, depending on the grammar system).

That is why you say:

• 上手です = is skillful / is good at
• 上手な人 = a skillful person

Notice:

• before a noun: 上手な
• at the end of a sentence: 上手です

So yes, you can think of it as an adjective for learning purposes, but specifically it is a na-adjective.

It can mean different but related things depending on context:

• cooking as an activity
• cuisine / dishes / food preparation

In 料理が上手です, it means good at cooking.

So here it refers to the skill or area of cooking, not just food itself.

Japanese does not always match English word-for-word.

English says:

• good at cooking

Japanese says:

• 料理が上手

Literally, this is closer to:

• cooking is skillful

But natural English translation is:

• good at cooking

So the idea of at is built into how 上手 works with が. You do not need a separate word meaning at here.

Not exactly, though the ideas are close.

• 私の姉は料理が上手です。 means My older sister is good at cooking.
• It describes her ability or skill.

If you want to say She cooks well, focusing more on the manner of the action, you might say something like:

• 私の姉は料理が上手です。 = She is skilled at cooking.
• 私の姉は上手に料理します。 = She cooks skillfully / well.

The original sentence is about her general skill, not just one action.

Yes, very often.

Japanese frequently omits information that is already clear from context. If it is already understood whose sister is being discussed, you could simply say:

• 姉は料理が上手です。

This would often still be understood as:

• My older sister is good at cooking.

Japanese uses context much more than English does.

Yes. です makes the sentence polite and complete.

Compare:

• 私の姉は料理が上手です。 = polite
• 私の姉は料理が上手だ。 = plain
• 私の姉は料理が上手。 = casual, often in conversation

So です does not add new meaning like is in a one-to-one way. It mainly gives polite sentence ending style here.

The sentence is:

• 私 / の / 姉 / は / 料理 / が / 上手 / です

A helpful breakdown is:

• 私の姉 = my older sister
• は = topic marker
• 料理 = cooking
• が = marks what she is good at
• 上手です = is good / skillful

So the structure is roughly:

• [Topic] は [thing] が 上手です

In natural English:

• My older sister is good at cooking.

Japanese often puts the main descriptive word near the end of the sentence.

Yes, often you can, but the nuance is a little different.

• 上手 = skillful, good at
• 得意 = strong point, something one is good at, often with a nuance of confidence or specialty

So:

• 私の姉は料理が上手です。 = My sister is good at cooking.
• 私の姉は料理が得意です。 = Cooking is one of my sister’s strong points.

Both are natural, but 上手 focuses more directly on skill.

Because Japanese usually does not mark these things the same way English does.

• 姉 already means older sister, so gender and relative age are built into the word.
• There is no need for older separately.
• There is also no plural marking because the sentence is about one sister.

Japanese often encodes family relationships more specifically than English:

• 姉 = older sister
• 妹 = younger sister
• 兄 = older brother
• 弟 = younger brother

So 姉 already gives more specific information than English sister.