Learn Hardcore Portuguese (Portugal): A minha prima quer estudar medicina para ser cientista num hospital público. - My cousin wants to study medicine to be a scientist in a public hospital.

In European Portuguese it sounds more marked: poetic, emphatic, or sometimes old‑fashioned.
In Brazilian Portuguese, dropping the article (minha prima) is much more common and sounds neutral.

So for everyday European Portuguese, prefer A minha prima.

Is prima specifically “female cousin”? How would I say “male cousin”?

Yes:

prima = female cousin
primo = male cousin

The article and possessive agree with gender:

A minha prima = my (female) cousin
O meu primo = my (male) cousin

The -a ending usually marks feminine; -o usually marks masculine.

Why is it quer and not querer or something like queres?

Querer is the infinitive, “to want”. It must be conjugated to agree with the subject:

eu quero – I want
tu queres – you (singular, informal) want
ele / ela / você quer – he / she / you want
nós queremos – we want
eles / elas / vocês querem – they / you (plural) want

The subject here is a minha prima = ela (she), so the correct form is quer:

A minha prima quer… = My cousin (she) wants…

Why is it quer estudar with estudar (infinitive) and not something like quer estuda?

After verbs like querer (to want), poder (can), conseguir (manage to), etc., Portuguese uses the infinitive:

quer estudar – wants to study
quer comer – wants to eat
pode sair – can go out

You never conjugate the second verb in this structure. So quer estudar is correct; quer estuda is wrong.

Why is it medicina and not something like remédio or medicamento?

Different words, different uses:

medicina = medicine as a field of study / profession / science
- estudar medicina = study medicine (at university)
remédio / medicamento = a (specific) medicine, a drug / pill
- tomar um remédio = take a medicine

In this sentence she wants to study the academic subject, so medicina is the right word.

Could I write Medicina with a capital letter, like in English for some school subjects?

In Portuguese, names of school/university subjects are normally written in lower case:

estudar medicina, história, biologia, matemática

Capitals are reserved for proper names (Portuguese, European History II, etc.) or official course titles. So in this sentence medicina should be lowercase.

Why is it para ser cientista for “to become a scientist”? Can’t I say para tornar-se cientista?

To express a long‑term profession or role as a goal, para ser + profession/role is the most natural and common:

estuda medicina para ser médico – studies medicine to become a doctor
treina muito para ser atleta profissional – trains a lot to become a professional athlete

You can say para tornar-se cientista/para se tornar cientista, but it:

sounds more formal or literary, and
focuses more on the process of becoming.

Everyday European Portuguese would strongly prefer para ser cientista here.

Why is there no uma before cientista? Why not para ser uma cientista?

With professions and long‑term roles, Portuguese usually drops um/uma after ser:

Ela é cientista. – She is a scientist.
Quero ser médico. – I want to be a doctor.
Ele vai ser engenheiro. – He is going to be an engineer.

So para ser cientista is the default, natural form.

Using para ser uma cientista is not wrong, but it tends to:

sound more specific (a scientist, maybe of a certain kind), or
add a bit of emphasis / nuance (e.g. “to be a real scientist”).

In a neutral sentence about career plans, para ser cientista is best.

Is cientista feminine here? How do I know the gender if it ends in -a but can also be masculine?

Cientista is one of those nouns that has the same form for masculine and feminine:

o cientista – the (male) scientist
a cientista – the (female) scientist

Here, we know it is feminine because the person is a minha prima (female cousin). If there were an article, it would be uma cientista.

So:

Masculine: o meu primo quer ser cientista num hospital público.
Feminine: a minha prima quer ser cientista num hospital público.

What exactly does num mean in num hospital público?

Num is a contraction:

em + um = num

So num hospital literally means “in a hospital” / “at a hospital”.

Similarly:

em + uma = numa – numa escola (“in a school”)
em + o = no – no hospital (“in the hospital”)
em + a = na – na escola (“in the school”)

In speech and informal writing, num is very common in both European and Brazilian Portuguese. In more careful or formal writing you might also see em um hospital, but num hospital is very natural.

What is the difference between num hospital público and no hospital público?

num hospital público = in a public hospital (any such hospital, not specified)
no hospital público = in the public hospital (a particular one, already known from context)

So:

Indefinite: Ela trabalha num hospital público. – She works in a (some) public hospital.
Definite: Ela trabalha no hospital público da cidade. – She works in the city’s public hospital.

Why is it hospital público and not público hospital, like “public hospital” in English?

In Portuguese, adjectives normally come after the noun:

hospital público – public hospital
carro vermelho – red car
livro interessante – interesting book

Placing the adjective before the noun (público hospital) is almost always wrong or would sound very odd and poetic.

So the normal order is hospital público: noun first, adjective after.

Why does público end in -o? Is it agreeing with hospital?

Yes. Adjectives in Portuguese must agree in gender and number with the noun:

hospital is masculine singular
so the adjective must be masculine singular: público

Examples of agreement:

um hospital público – a public hospital
dois hospitais públicos – two public hospitals
uma escola pública – a public school (feminine singular)
duas escolas públicas – two public schools (feminine plural)

The endings -o / -a / -os / -as mark that agreement.

Why is ser used here (para ser cientista) and not estar?

Ser is used for:

permanent characteristics,
identity,
professions.

Being a scientist is treated as a stable identity/profession, so:

para ser cientista – to be/become a scientist (profession/identity)

Estar is for temporary states or locations:

estar cansado – to be tired (right now)
estar no hospital – to be in the hospital (location)

Using estar cientista would be ungrammatical in this sense.