Learn Hardcore Portuguese (Portugal): Não, hoje não quero ir ao mercado. - No, today I don't want to go to the market.

The first Não, = No, as a short answer to a previous question (e.g. Queres ir ao mercado? – Não, …).
The second não is the normal sentence negation: (Eu) não quero = I don’t want.

So the structure is essentially:

Não, = No,
hoje não quero ir ao mercado. = today I don’t want to go to the market.

Can I remove the first Não and just say Hoje não quero ir ao mercado?

Yes.

Hoje não quero ir ao mercado. = Today I don’t want to go to the market.

Here you are not explicitly answering “No”; you are just stating your unwillingness.
With the initial Não, you explicitly answer a yes/no question first and then give the explanation:

Queres ir ao mercado?
Não, hoje não quero ir ao mercado.

Why is there no eu? Could I say Não, hoje eu não quero ir ao mercado?

Portuguese is a pro‑drop language: subject pronouns are often omitted because the verb ending already shows the subject.

(Eu) quero = I want
(Tu) queres = you want
(Ele / ela / você) quer = he / she / you want

So Não, hoje não quero ir ao mercado naturally means No, today I don’t want to go to the market with eu understood.

You can add eu for emphasis or contrast:

Não, hoje eu não quero ir ao mercado, mas tu podes ir.
No, today *I don’t want to go to the market, but you can go.*

Without a reason to emphasise I, most Europeans would leave eu out here.

Why is it quero and not quer or querer?

Quero is the first person singular (eu) form of querer in the present tense:

eu quero – I want
tu queres – you (singular, informal) want
ele / ela / você quer – he / she / you (formal) wants

So:

(Eu) não quero = I don’t want
(Ele) não quer = He doesn’t want

In your sentence, the implied subject is eu, so the correct form is quero.

Can I say Não quero ir ao mercado hoje instead of Não, hoje não quero ir ao mercado? Is there a difference?

Yes, you can say:

Não quero ir ao mercado hoje.

The meaning is basically the same; it’s a matter of word order and emphasis:

Hoje não quero ir ao mercado. – Slight emphasis on today (as opposed to another day).
Não quero ir ao mercado hoje. – Slight emphasis on not wanting to go to the market today (neutral, very common).
Não, hoje não quero ir ao mercado. – Explicit answer “No,” plus the emphasis on today.

All are grammatically correct in European Portuguese.

What does ao mean in ir ao mercado?

Ao is a contraction:

a (preposition to) + o (masculine singular article the) → ao

So:

ir a o mercado → ir ao mercado = to go to the market

You must use the contraction in normal speech and writing; ir a o mercado sounds wrong/foreign.

What is the difference between ir ao mercado, ir para o mercado, and ir no mercado?

In European Portuguese:

ir ao mercado
- Literally: to go to the market
- Most natural and standard for going to the market as a destination.
ir para o mercado
- Also to go to the market, often with a slight focus on heading there / staying there for some purpose.
- In this context, it usually works similarly to ir ao mercado, but ao mercado is more common and neutral.
ir no mercado
- Literally: to go in/at the market
- This is not used for “go to the market”.
- You would hear no mercado to mean in/at the market:
  - Estou no mercado. = I’m at the market.

So for your sentence, the natural European Portuguese choice is ir ao mercado.

Can this sentence also be used in Brazilian Portuguese, or is it only European?

The sentence is perfectly correct in both European and Brazilian Portuguese:

Não, hoje não quero ir ao mercado.

Both varieties understand and use ir ao mercado. In everyday conversation Brazilians might also say ir no mercado in some regions when they mean “go to the supermarket”, but that is regional/colloquial and not standard European Portuguese.

Pronunciation and intonation will differ, but the grammar and wording are fine for both.

Why is there a comma after Não?

The comma separates:

the short answer: Não, = No,
from the full sentence: hoje não quero ir ao mercado.

In English you’d also write: No, today I don’t want to go to the market.

Without the comma, it would be harder to see that the first Não is a separate, stand‑alone reply.

How do you pronounce não, hoje, and mercado in European Portuguese?

Approximate European Portuguese pronunciation:

não – nasal sound; roughly like “nah-oo” blended together. IPA: /nɐ̃w̃/.
hoje – starts with an h that is silent; the j is like the s in “measure”. Roughly: “OH-zh(ɨ)”. IPA: /ˈoʒɨ/.
mercado – r is a tapped r (like Spanish pero), final o is fairly closed. Roughly: “mər-KAH-doo”. IPA: /mɨɾˈkaðu/ (or /mɨɾˈkadu/ depending on accent).

Spoken slowly:

Não, hoje não quero ir ao mercado.
/nɐ̃w̃ ˈoʒɨ nɐ̃w̃ ˈkɛɾu iɾ aw mɨɾˈkaðu/

Does Não, hoje não quero ir ao mercado sound rude in Portuguese?

By itself, it is direct but not necessarily rude. Context and tone of voice matter a lot.

To sound softer/more polite in European Portuguese, people often choose alternatives like:

Hoje não me apetece ir ao mercado.
Today I don’t feel like going to the market.
Hoje preferia não ir ao mercado.
Today I’d rather not go to the market.
Hoje não queria ir ao mercado.
Today I wouldn’t like to go to the market.

Your original sentence is fine among friends or family; with strangers or in more formal situations, softer phrasing can be more polite.

Does mercado always mean the same as English market or supermarket?

Mercado is flexible and can mean:

A traditional market (with stalls): mercado municipal = municipal market.
A supermarket, in everyday speech, depending on context:
- Vou ao mercado. – Often understood as I’m going to the supermarket.
The abstract “market” (economics): o mercado de trabalho = the job market.

In your sentence, ir ao mercado is easily understood as go to the market / supermarket, depending on what’s normal in that context.