Learn Hardcore Portuguese (Portugal): Muitos alunos querem estudar no estrangeiro depois da faculdade. - Many students want to study abroad after college.

muito dinheiro = much money
muitas pessoas = many people
muito inteligente = very intelligent (here muito is an adverb, so it does not change form)

Is there a difference between alunos and estudantes?

Both can mean students, but there are some tendencies:

aluno
- very common and neutral
- used for school pupils and university students
- in European Portuguese, people often say alunos da faculdade for university students
estudante
- also common
- slightly more formal or generic
- often used in written language, or when we do not care about the education level

In your sentence, Muitos alunos querem estudar... could easily also be Muitos estudantes querem estudar... without a big change in meaning. The context would tell you whether they are school students or university students.

How does querem estudar work? Why are there two verbs and only one is conjugated?

Portuguese often uses a conjugated verb followed by an infinitive, similar to English want to study, need to go, etc.

querem is the present tense of querer (they want)
estudar is the infinitive (to study)

So querem estudar literally matches English (they) want to study, but Portuguese does not use a separate word for to here:

Eles querem estudar. = They want to study.
Ela quer viajar. = She wants to travel.

Some other verbs that behave similarly:

precisar de estudar (to need to study — note the de)
começar a estudar (to start studying — note the a)

With querer, you just use querer + infinitive, without a preposition:

querer estudar, querer viajar, querer trabalhar, etc.

Why do we say no estrangeiro and not something like em estrangeiro?

No is a contraction:

no = em + o (in/on the)

So no estrangeiro literally means in the foreign (place), which corresponds idiomatically to English abroad.

You cannot say em estrangeiro, because here estrangeiro is a noun (like the foreign country / foreign world) and in Portuguese nouns almost always need an article (o / a / os / as) in this kind of context.

Structure:

em + o estrangeiro → no estrangeiro = abroad

Compare with estrangeiro as an adjective:

um aluno estrangeiro = a foreign student
uma empresa estrangeira = a foreign company

As a noun, with article: o estrangeiro = abroad, foreign countries in general.

Could we also say no exterior instead of no estrangeiro? Is there a difference?

Yes, no exterior is also used and understood as abroad, especially in more formal contexts or in writing.

Nuances (in European Portuguese):

no estrangeiro
- very common in everyday speech
- focuses on the idea of foreign countries in general
no exterior
- a bit more formal or neutral
- often used in official or news contexts: trabalhar no exterior, portugueses a viver no exterior

In most situations, you can swap them without changing the basic meaning:

Muitos alunos querem estudar no estrangeiro.
Muitos alunos querem estudar no exterior.

Both are natural.

Why is it depois da faculdade and not just depois faculdade or depois a faculdade?

Depois normally takes de before a noun:

depois de = after

When the noun has a definite article, de + a (or de + o) contracts:

de + a faculdade → da faculdade
de + o trabalho → do trabalho

So:

depois de
- a faculdade → depois da faculdade
depois de
- o trabalho → depois do trabalho

You cannot say depois faculdade or depois a faculdade; the preposition de is required.

Notice the patterns:

depois da faculdade = after college / university
depois do trabalho = after work
depois de estudar = after studying (here it is depois de
- verb in the infinitive, so no article)

What exactly does faculdade mean in European Portuguese? Is it faculty, college, or university?

In European Portuguese, faculdade usually refers to:

a higher-education unit, typically a faculty/college belonging to a university:
- Faculdade de Letras, Faculdade de Medicina, etc.
colloquially, people often use a faculdade to mean university studies in general, similar to English college or uni.

So depois da faculdade is best understood in English as:

after college or
after university

It does not mean after the ability (the English word faculty in the sense of mental faculties) here.

For that sense, Portuguese uses other words, like capacidade.

Why do we use articles in Portuguese (no estrangeiro, da faculdade) when in English we say abroad, after college without the?

Portuguese uses definite articles much more frequently than English, even for general concepts.

In your sentence:

no estrangeiro = em + o estrangeiro
- literally in the foreign (place)
- idiomatically: abroad
da faculdade = de + a faculdade
- literally of/from the college/faculty
- idiomatically in this context: after college / after university

English often drops the with general or abstract places/activities:

at school, in hospital, at university, in prison, after work, after college

Portuguese usually keeps the article:

na escola (in the school / at school)
no hospital (in the hospital / in hospital)
na universidade / na faculdade (at university / in college)

So the presence of o / a and contractions like no, da is normal and often has no direct English equivalent.

Why is it muitos alunos (masculine) and not muitos alunas or muitas alunas?

In Portuguese, when you are talking about:

a group of only men/boys → masculine plural
a group of only women/girls → feminine plural
a mixed or unspecified group → default to masculine plural

So:

muitos alunos
- could mean many male students
- or many students in general (mixed or gender-unknown group)
muitas alunas
- specifically many female students

In this sentence, muitos alunos is generic: many students. The gender is not specified; masculine plural is just the default for mixed or unknown groups.

Why is querem in the present tense? How would I say wanted to study or will want to study?

Querem is the 3rd person plural present tense of querer:

eles / elas querem = they want

So:

Muitos alunos querem estudar... = Many students want to study...

Other tenses:

Muitos alunos queriam estudar no estrangeiro depois da faculdade.
- many students wanted to study abroad after college (imperfect past, often background/ongoing in the past)
Muitos alunos quiseram estudar no estrangeiro depois da faculdade.
- many students wanted (and typically tried/decided at a specific time) to study abroad (simple past with a more completed, event-like feeling)
Muitos alunos vão querer estudar no estrangeiro depois da faculdade.
- many students will want to study abroad

The structure querer + infinitive stays the same; only querer changes form.

How do you pronounce the tricky parts in European Portuguese, like muitos, estrangeiro, and depois?

Approximate European Portuguese pronunciation (very rough English-like guides):

muitos
- IPA: /ˈmũjtuʃ/
- roughly: MOYN-toosh
- nasal ui sound, and final s sounds like sh
alunos
- IPA: /ɐˈlunuʃ/
- roughly: uh-LOO-noosh
querem
- IPA: /ˈkɛɾẽj̃/ (varies by region)
- roughly: KEH-reng (with a tapped r, like a very quick d)
no
- IPA: /nu/
- like noo in noon
estrangeiro
- IPA: /ʃtɾɐ̃ˈʒɐjɾu/
- roughly: shtruhn-ZHAY-roo
- initial es- before t becomes sh-like
- gei is like zhay (the zh in French jour)
- the r is tapped
depois
- IPA: /dɨˈpojʃ/
- roughly: dih-POYSH
- the final s again sounds like sh
faculdade
- IPA: /fɐkulˈdad(ɨ)/
- roughly: fuh-cool-DAHd(ɨ)
- stress on DA

These are approximations to help your ear; listening to native speakers is the best way to get used to European Portuguese sounds.