Learn Hardcore Japanese: sensyuu ha hazimete imouto ni ryouri wo osieta. imouto ha sugu zyouzu ni natta. - Last week I taught my younger sister to cook for the first time. She got good quickly.

Here, 妹に is the recipient (to my younger sister).
料理を is the direct object (the thing being taught). You can swap their order before the verb without changing meaning: 料理を妹に教えた is also fine.

What exactly does 初めて modify, and where can it go?

初めて is an adverb meaning “for the first time,” and it modifies the action 教えた. Natural placements include:

先週、初めて妹に料理を教えた。
先週は初めて妹に料理を教えた。
妹に初めて料理を教えた（先週）。 Avoid pairing 初めて directly with a time word like 初めて先週; instead, keep 初めて near the verb phrase or place it early in the sentence with a comma.

Why is 先週 followed by は? Could it be just 先週 or 先週に?

Time words like 先週 often appear without a particle: 先週、… is the neutral, common choice. Using 先週は makes “last week” the topic, often adding a light contrast like “as for last week.” 先週に is possible but comparatively uncommon in simple past statements like this; it’s more typical in formal writing or specific structures (e.g., 先週に起こった出来事).

Why isn’t there an explicit subject like “I”?

Japanese frequently omits subjects when they’re clear from context. In a first-person narrative, the understood subject is typically the speaker. If you want to state it, you can say:

私は先週、初めて妹に料理を教えた。 Using 私が instead stresses that it was specifically you who did it.

In the second sentence, why 妹は and not 妹が?

は marks the topic. Since 妹 was already introduced in the first sentence, using 妹は to continue talking about her is natural. 妹がすぐ上手になった is also grammatical, but it can sound like you’re introducing this as new, focused information or emphasizing that it was she (as opposed to someone else) who improved.

Why is it 上手になった? Could it be 上手になった or うまくなった?

上手 is a na-adjective. With なる, na-adjectives take に: 上手になる / 上手になった (no space in normal writing; the original spacing just reflects word boundaries).
うまい is an i-adjective, so with なる it becomes うまくなる / うまくなった. Nuance:
上手 sounds a bit more neutral/objective.
うまい is more colloquial and can also mean “tasty” in other contexts, but with なる it clearly means “to become skillful.”

What does すぐ add? Is すぐに different?

すぐ means “immediately/soon.” すぐに is essentially the same; に can add a slightly more formal or emphatic feel. Both are fine with なる here.

Do I need to say 私の妹? And should I use 妹さん?

For your own family members, it’s natural to say simply 妹 (no 私の, no さん). Use 私の妹 if you need to avoid ambiguity. Use 妹さん when referring to someone else’s younger sister (honorific).

Is 料理を教える the same as “teach how to cook”? Would 料理の作り方を教える be better?

料理を教える is a natural, concise way to say “teach cooking” and is fine. If you want to be explicit about the method, 料理の作り方を教える or 料理の仕方を教える both work. Don’t say 料理を作り方; it needs の: 料理の作り方.

Can I drop particles like を or に in casual speech?

In casual conversation, を is sometimes dropped: 料理(を)教えた. However, dropping に (the recipient marker) is risky because it can create ambiguity. For clarity—especially as a learner—keep both に and を.

How flexible is the word order before the verb?

Fairly flexible. All of these are natural, with slight differences in emphasis:

先週、初めて妹に料理を教えた。
先週、妹に初めて料理を教えた。
妹に先週、初めて料理を教えた。 Elements before the verb can be rearranged for focus, but avoid unnatural pairings like 初めて先週.

How do I make the sentences polite?

Use the -ます form:

先週は初めて妹に料理を教えました。妹はすぐ上手になりました。 Keep the style consistent (don’t mix casual and polite in the same context unless you have a reason).

What’s the difference between 教える and 習う／教わる? Could one of those fit here?

教える = to teach (teacher’s perspective): 妹に料理を教えた.
習う/教わる = to learn / be taught (learner’s perspective): 妹は私に料理を習った／私に料理を教わった. Use the one that matches the perspective you want.

Could I omit 妹 in the second sentence and just say すぐ上手になった?

You could, but it becomes ambiguous about who improved. Reintroducing 妹は sets the topic clearly and reads more naturally. Pronouns like 彼女 are generally avoided when a specific noun (妹) is available.