No quiero comprar nada hoy.

Because Spanish often leaves out subject pronouns when they are clear from the verb form.

• Quiero already means I want
• So No quiero comprar nada hoy is perfectly complete without yo

You can add yo for emphasis or contrast:

• Yo no quiero comprar nada hoy, pero ella sí.
  The idea is I don’t want to buy anything today, but she does.

That is because querer is a stem-changing verb.

In the present tense, the e in the stem changes to ie in most forms:

• yo quiero
• tú quieres
• él/ella quiere
• nosotros queremos
• vosotros queréis
• ellos quieren

So quiero is the correct I form of querer in the present tense.

Because after a conjugated verb like querer, the next verb usually stays in the infinitive.

So Spanish works like this:

• quiero comprar = I want to buy
• quiero comer = I want to eat
• quiero salir = I want to go out

Only the first verb is conjugated here. The second one stays in its base form.

Yes, but Spanish normally uses this kind of structure. It is called negative concord.

In Spanish, it is very common to have:

• no before the verb
• another negative word after the verb, such as nada, nadie, nunca

So:

• No quiero comprar nada hoy

literally looks like I don’t want to buy nothing today, but it actually means I don’t want to buy anything today.

This is normal and correct Spanish.

Here it is best translated as anything because it appears in a negative sentence.

Compare:

• No quiero comprar nada hoy = I do not want to buy anything today
• Nada es fácil = Nothing is easy

So nada can correspond to either nothing or anything in English, depending on the structure.

After no, Spanish normally prefers negative words, not positive-indefinite words.

So Spanish says:

• No quiero comprar nada hoy

not usually:

• No quiero comprar algo hoy

The version with algo sounds odd in most normal situations.
After a negative verb, Spanish generally uses:

• nada = anything / nothing
• nadie = anybody / nobody
• nunca = ever / never
• ningún / ninguna = any / no

Yes. Hoy is flexible.

These are all possible:

• No quiero comprar nada hoy
• Hoy no quiero comprar nada

Both are natural. The difference is mainly focus:

• No quiero comprar nada hoy → neutral, very common
• Hoy no quiero comprar nada → puts more emphasis on today

So the original sentence is natural, but it is not the only possible word order.

Yes, but the meaning becomes broader.

• No quiero comprar nada hoy = I don’t want to buy anything today
• No quiero nada hoy = I don’t want anything today

Without comprar, the sentence is no longer specifically about buying. It could mean you do not want anything at all.

A simple English-friendly approximation is:

• noh KYE-ro kom-PRAR NA-dha oy

A few helpful points:

• hoy: the h is silent, so it sounds like oy
• qu in quiero sounds like k
• r in quiero is a light Spanish r
• comprar has the stress on the last syllable: com-PRAR
• in Spain, the d in nada is often softer than English d

You do not need a perfect accent at first; the most important thing is getting the stress right:

• quie-ro
• com-prar
• na-da
• hoy

It is neutral and works in both formal and informal situations.

You could say it:

• to a friend
• to a shop assistant
• to family
• in a normal conversation

There is nothing especially casual or especially formal about it. It is just standard everyday Spanish.