Θέλω να κάνω αίτηση για το πρόγραμμα ανταλλαγής φοιτητών.

After θέλω (I want), Greek normally uses να plus a verb:

Θέλω να κάνω... = I want to do/make...

So in this kind of sentence, να often corresponds to English to, but grammatically it is not an infinitive marker in the same way English uses to. It introduces a subjunctive-type verb form, which is the normal pattern after verbs like θέλω.

A very common structure is:

θέλω να + verb
= want to + verb

For example:

• Θέλω να φύγω = I want to leave
• Θέλω να μάθω ελληνικά = I want to learn Greek

Yes. Κάνω αίτηση is a very common and natural way to say apply.

Literally, it means something like:

• κάνω = do / make
• αίτηση = application

So the phrase literally means make an application, but in normal English we usually translate it as apply.

A more formal alternative is:

υποβάλλω αίτηση = submit an application

But for everyday use, κάνω αίτηση is extremely common.

Greek often leaves out the indefinite article in fixed expressions like this.

So:

κάνω αίτηση
naturally means
apply / make an application

You can say κάνω μια αίτηση if you want to emphasize one application, but it is not necessary here.

Compare:

• Κάνω αίτηση για το πρόγραμμα = I’m applying for the program
• Κάνω μια αίτηση σήμερα = I’m submitting one application today

So the version without μια is perfectly normal.

Here για means for.

So:

κάνω αίτηση για κάτι
= apply for something

That is the standard pattern here.

Examples:

• κάνω αίτηση για υποτροφία = apply for a scholarship
• κάνω αίτηση για δουλειά = apply for a job
• κάνω αίτηση για το πρόγραμμα = apply for the program

You may also see σε with the organization you apply to:

• Κάνω αίτηση στο πανεπιστήμιο = I apply to the university
• Κάνω αίτηση στο πανεπιστήμιο για υποτροφία = I apply to the university for a scholarship

So για tells you what you are applying for.

It is in the accusative, because για takes the accusative case.

So in:

για το πρόγραμμα

the noun phrase is accusative.

A useful thing to know is that neuter nouns like πρόγραμμα have the same form in the nominative and accusative singular:

• nominative: το πρόγραμμα
• accusative: το πρόγραμμα

So even though the form looks the same, its role here is accusative because it follows για.

This is a very typical Greek noun chain using the genitive.

Break it down like this:

• το πρόγραμμα = the program
• ανταλλαγής = of exchange
• φοιτητών = of students

So literally:

το πρόγραμμα ανταλλαγής φοιτητών
= the program of exchange of students

In natural English, we usually compress that to:

the student exchange program

So:

• ανταλλαγής is the genitive singular of ανταλλαγή (exchange)
• φοιτητών is the genitive plural of φοιτητής (student)

Greek often expresses noun relationships this way, where English might use adjective-like nouns instead.

Because the phrase refers to:

• one exchange as a concept: ανταλλαγής = of exchange
• many students: φοιτητών = of students

So the structure is:

program → of exchange → of students

It would sound strange to make exchange plural here, because the phrase is naming the type of program, not counting multiple exchanges.

English does something similar conceptually in student exchange program: exchange is singular, students is understood as the group involved.

Usually yes, if you mean a specific program.

για το πρόγραμμα ανταλλαγής φοιτητών
means
for the student exchange program

That sounds like a particular, known program.

If you remove the article:

για πρόγραμμα ανταλλαγής φοιτητών

it becomes more indefinite, like for a student exchange program or just a more general mention. That can be possible in some contexts, but in this sentence the definite article is the most natural choice.

Usually φοιτητής means a university or college student.

So:

• φοιτητής / φοιτητών = university student / students
• μαθητής / μαθητών = school pupil / pupils

That means πρόγραμμα ανταλλαγής φοιτητών normally suggests a university student exchange program, not a school exchange.

Greek word order is more flexible than English word order, but the original sentence is the most neutral and natural.

Standard order:

Θέλω να κάνω αίτηση για το πρόγραμμα ανταλλαγής φοιτητών.

You can move parts around for emphasis. For example:

Για το πρόγραμμα ανταλλαγής φοιτητών θέλω να κάνω αίτηση.

That puts extra focus on for the student exchange program.

So the sentence is not completely fixed, but the original version is the safest one for a learner.

A very common polite version is:

Θα ήθελα να κάνω αίτηση για το πρόγραμμα ανταλλαγής φοιτητών.

This means:

I would like to apply for the student exchange program.

Compare the tone:

• Θέλω να... = I want to...
  More direct.
• Θα ήθελα να... = I would like to...
  More polite and softer.

So if you are speaking to an office, administrator, or professor, Θα ήθελα να... is often better.

The accent marks show which syllable is stressed.

Stress is important in Greek, and moving it can make a word sound wrong or even change the word.

In this sentence:

• Θέλω → stress on the first syllable
• κάνω → stress on the first syllable
• αίτηση → stress on the first syllable
• πρόγραμμα → stress on the first syllable
• ανταλλαγής → stress on the last syllable
• φοιτητών → stress on the last syllable

A rough pronunciation guide is:

• Θέλω = THE-lo
• να = na
• κάνω = KA-no
• αίτηση = E-ti-si
• για = ya
• το = to
• πρόγραμμα = PRO-gra-ma
• ανταλλαγής = an-da-la-YIS
• φοιτητών = fi-ti-TON

Yes.

υποβάλλω αίτηση is more formal and more literally means submit an application.

So:

• Θέλω να κάνω αίτηση... = natural, common, everyday
• Θέλω να υποβάλω αίτηση... = more formal, administrative style

Both are correct. If you are speaking casually, κάνω αίτηση is usually the easier and more natural choice.