Mi prima quiere estudiar repostería.

Because mi here is a possessive adjective meaning my.

• mi = my
• mí = me after a preposition, as in para mí (for me)

So in Mi prima, mi has no accent mark.

No. Mi stays the same for both masculine and feminine singular nouns.

• mi primo = my male cousin
• mi prima = my female cousin

Only the noun changes for gender, not mi.
In the plural, it becomes mis:

• mis primos
• mis primas

Prima means female cousin. It is the feminine form of primo.

• primo = male cousin
• prima = female cousin

So Spanish makes the cousin’s gender explicit here, unlike English, which just says cousin.

Because quiere is the conjugated form of the verb querer for él/ella/usted in the present tense.

The verb querer means to want or sometimes to love depending on context.

Present tense:

• yo quiero
• tú quieres
• él/ella/usted quiere
• nosotros/nosotras queremos
• vosotros/vosotras queréis
• ellos/ellas/ustedes quieren

Since the subject is mi prima = she, the correct form is quiere.

Because querer is a stem-changing verb in the present tense.

Its stem changes:

• quer- → quier-

So:

• yo quiero
• tú quieres
• él/ella quiere
• ellos quieren

But the change does not happen in nosotros or vosotros:

• nosotros queremos
• vosotros queréis

This is a very common pattern in Spanish.

After a conjugated verb like quiere, the next verb stays in the infinitive.

So:

• quiere estudiar = wants to study
• quiere comer = wants to eat
• quiere aprender = wants to learn

In Spanish, you do not need a separate word for to before the second verb in this kind of structure. The infinitive itself already gives that idea.

Because Spanish often uses a structure that is simply:

conjugated verb + infinitive

So:

• quiere estudiar = wants to study
• necesita trabajar = needs to work
• prefiere esperar = prefers to wait

English uses to before the second verb, but Spanish usually does not in this pattern.

Because when Spanish talks about studying a subject in a general sense, it often does not use an article.

So:

• estudiar repostería
• estudiar medicina
• estudiar derecho
• estudiar música

This is similar to English, where we also usually say study medicine, not study the medicine.

Here, repostería is a noun referring to the field or subject of pastry-making / confectionery / baking sweets.

In this sentence, it works like the thing she wants to study.

So grammatically:

• Mi prima = subject
• quiere = conjugated verb
• estudiar = infinitive
• repostería = object/complement of estudiar

The accent mark shows where the stress goes: re-po-ste-RÍ-a.

Without going too deep into spelling rules, the written accent is needed to show the correct pronunciation and stress pattern.

It also helps indicate that -ía is pronounced clearly, with the stress on rí.

Because Spanish often omits subject pronouns when the verb form already makes the subject clear.

Instead of saying:

• Ella quiere estudiar repostería

Spanish very naturally says:

• Mi prima quiere estudiar repostería

And even if the subject were already known, Spanish could simply say:

• Quiere estudiar repostería

This is normal because Spanish verb endings carry a lot of information.

Not in this sentence. Here querer + infinitive clearly means to want to do something.

• quiere estudiar = wants to study

When querer means to love / care for someone, it usually appears with a direct object, often a person:

• Quiere a su prima = He/She loves his/her cousin

So the structure changes the meaning:

• querer + infinitive = to want to do
• querer + noun/person = often to love / want

The normal, neutral order is:

Mi prima quiere estudiar repostería.

That said, Spanish can sometimes move parts around for emphasis, but this is the most natural basic order for a learner:

subject + verb + infinitive + object

So this sentence is a very standard model to copy.