Learn Hardcore Russian: Мне надо купить морковь на рынке.

Because надо is often used in an impersonal construction in Russian. Instead of saying Я (I) as the subject, Russian commonly expresses “I need to…” as “To me it is necessary…”.
So Мне is the dative case of я and means to me / for me, giving: Мне надо + infinitive = I need to + verb.

What exactly is надо grammatically?

Надо is a predicative word (often treated like an impersonal modal) meaning it is necessary / need to. It typically combines with:

a person in the dative (мне, тебе, ему, etc.)
an infinitive (купить, сделать, идти, etc.)

Pattern: [Dative] + надо + infinitive.

Can I replace надо with нужно? Is there a difference?

Yes, you can usually replace надо with нужно with little change in meaning:

Мне надо купить морковь.
Мне нужно купить морковь.

Typical nuance:

надо often feels a bit more conversational/direct (“gotta / need to”)
нужно often feels slightly more neutral or “standard,” sometimes a bit more formal

In many everyday situations, they’re interchangeable.

Why is купить used (perfective) instead of покупать (imperfective)?

Купить is perfective and focuses on a single completed result: you will successfully buy the carrots.
You’d use покупать (imperfective) for:

repeated/habitual actions: Мне надо покупать морковь каждую неделю. (I need to buy carrots every week.)
process/ongoing emphasis (less common with надо in this exact meaning, but possible in context)

Here, the likely meaning is one конкретная покупка, so купить fits best.

What case is морковь here, and why doesn’t it look different?

Морковь is the direct object of купить, so it’s in the accusative case.

But морковь is:

feminine
inanimate
a “soft sign” noun

For many feminine inanimate nouns, accusative singular = nominative singular, so it looks the same: морковь.

(Compare: genitive would be моркови.)

Could I say морковку instead of морковь?

Yes. Морковку is a common colloquial form (often felt as more informal and/or a bit diminutive).

купить морковь = neutral/standard
купить морковку = more conversational (“some carrot / a carrot,” often with a homier tone)

Both are natural.

Why is it на рынке and not в рынке?

Russian uses на with many places that are thought of as open areas, venues, or “activity locations,” including:

на рынке (at the market)
на стадионе (at the stadium)
на концерте (at the concert)

В рынке would sound wrong for “at the market” in normal usage.

What case is рынке in на рынке?

After на meaning location (“where?”), Russian uses the prepositional case.
So:

рынок (dictionary form, nominative)
на рынке (prepositional)

Does на рынке mean “at the market” or “in the market”?

In English you might say “at the market” or “in the market,” but Russian standard phrasing is на рынке, and it covers the idea of being there to shop.

If you want to emphasize motion to the market, you’d say:

на рынок (accusative, “to the market”): Мне надо пойти на рынок.

Can the word order change? What’s the “default” order?

Word order is flexible. A common neutral order is exactly what you have:

Мне надо купить морковь на рынке.

But you can move parts for emphasis:

На рынке мне надо купить морковь. (emphasizes “at the market”)
Морковь мне надо купить на рынке. (emphasizes “carrots”)
Мне надо на рынке купить морковь. (slightly more spoken; emphasis can vary)

The meaning stays basically the same; the focus shifts.

Is the sentence missing a word like “I” or “to”?

No—Russian doesn’t need them here.

English needs: “I need to buy…”
Russian uses the infinitive directly after надо: надо купить (no separate word for “to”)

And я is not required because the “person” is already expressed by the dative мне.

How would I negate this sentence?

Put не before надо:

Мне не надо купить морковь на рынке.

This usually means “I don’t need to buy carrots at the market.”

If you mean “I must not / I’m not supposed to,” Russian often uses different wording depending on context (e.g., нельзя, не следует, etc.), but simple “don’t need to” is мне не надо.