Quiero enterarme de la nueva fecha de la reunión.

Because the verb is enterarse de, which is a pronominal verb. In this sentence, me is attached to the infinitive, giving enterarme.

• enterarse de algo = to find out about something / to learn about something
• quiero enterarme = I want to find out

After a conjugated verb like quiero, Spanish often attaches the pronoun to the infinitive:

• Quiero enterarme
• Necesito levantarme
• Prefiero irme

You could also say Me quiero enterar de..., and that is also correct.

Enterarse de means to find out about, to learn about, or to become aware of something.

In this sentence:

• Quiero enterarme de la nueva fecha de la reunión
  = I want to find out the new date of the meeting

A useful thing to remember is that enterarse usually suggests getting information you do not know yet. It is often close to find out in English.

Because enterarse normally goes with de.

The pattern is:

• enterarse de algo = to find out about something

So:

• enterarme de la nueva fecha
• se enteró de la noticia
• nos enteramos del problema

This de is required by the verb, not added randomly.

They do two different jobs:

1. The first de belongs to the verb enterarse de

   • enterarme de la nueva fecha

2. The second de shows possession or relation

   • la nueva fecha de la reunión = the new date of the meeting

So the full structure is:

• [Quiero] [enterarme de] [la nueva fecha de la reunión]

This is completely normal in Spanish.

This is a very common Spanish structure:

• querer + infinitive = to want to do something

So:

• quiero = I want
• enterarme = to find out

Together:

• Quiero enterarme = I want to find out

Other examples:

• Quiero comer = I want to eat
• Quiero salir = I want to go out
• Quiero enterarme de todo = I want to find out everything

Yes, you could, but the nuance is a little different.

• Quiero saber... = I want to know...
• Quiero enterarme de... = I want to find out about...

Saber focuses more on having the knowledge.
Enterarse de focuses more on getting the information.

So both are possible, but enterarme de sounds more like you do not know it yet and want to learn it.

In Spanish, adjectives usually come after the noun, but some adjectives often come before the noun, especially when they are common, descriptive, or emphasize a general quality.

Nuevo/nueva is one of those adjectives that often appears before the noun:

• la nueva fecha = the new date
• el nuevo coche = the new car

If you say la fecha nueva, it sounds less natural here and may suggest contrast, as if you are specifically distinguishing it from an old date in a more marked way.

For a learner, la nueva fecha is the normal choice.

Because it agrees with fecha, which is a feminine singular noun.

• la fecha = feminine singular
• nueva = feminine singular form of nuevo

Agreement in Spanish means articles and adjectives match the noun:

• el nuevo horario
• la nueva fecha
• los nuevos planes
• las nuevas fechas

Yes, reunión is feminine, so it takes la.

• la reunión = the meeting

Many nouns ending in -ión are feminine:

• la reunión
• la conversación
• la decisión
• la información

Also note the written accent in reunión. The stress falls on the last syllable.

Yes. With a conjugated verb plus an infinitive, Spanish often allows two positions for the pronoun:

• Quiero enterarme de la nueva fecha de la reunión
• Me quiero enterar de la nueva fecha de la reunión

Both are correct.

The version with the pronoun attached to the infinitive is very common and often feels a bit neater in writing.

Yes, it is natural and correct in Spain.

It sounds like:

• I want to find out the new date of the meeting
• I want to hear what the new meeting date is

Depending on context, native speakers might also say:

• Quiero saber la nueva fecha de la reunión
• Quiero enterarme de cuándo es la reunión al final
• Quiero saber cuál es la nueva fecha de la reunión

But your original sentence is absolutely fine and natural.

The whole phrase la nueva fecha de la reunión is what follows enterarme de.

Inside that phrase:

• la fecha = the main noun
• nueva = adjective describing fecha
• de la reunión = a phrase telling you which date

So:

• la nueva fecha de la reunión
  = the new date of the meeting

The core idea is that the speaker wants information about that date.

You would change quiero to queremos and enterarme to enterarnos:

• Queremos enterarnos de la nueva fecha de la reunión.

Breakdown:

• queremos = we want
• enterarnos = to find out / to get informed

Or with the pronoun before:

• Nos queremos enterar de la nueva fecha de la reunión.

Both are correct.