Learn Hardcore Greek: Η αδερφή μου τραγουδάει πολύ καλά και θέλει να γίνει μουσικός. - My sister sings very well and wants to become a musician.

Η is the definite article “the” in the feminine nominative singular. In Greek, you normally use the article with a noun even when there is a possessive pronoun like μου.

So Η αδερφή μου is literally “the sister of mine”, which corresponds to my sister.
Leaving out the article (Αδερφή μου τραγουδάει…) is possible in some contexts (for example as a form of address: Αδερφή μου, έλα εδώ! – My sister, come here!), but for a normal statement about a specific person, native speakers strongly prefer Η αδερφή μου.

Why is it αδερφή μου and not μου αδερφή, like in English my sister?

In Greek, the unstressed (enclitic) possessive pronouns like μου usually come after the noun they modify:

η αδερφή μου = my sister
ο αδερφός σου = your brother
το βιβλίο του = his book

You cannot put μου before the noun the way English does (my sister). The “normal” order is article + noun + possessive pronoun: η αδερφή μου.

If you want to emphasize possession, you can use stressed forms like δικιά μου:

η δικιά μου αδερφή = my own sister / my sister (as opposed to someone else’s)

But μου αδερφή is not correct.

I’ve seen αδελφή instead of αδερφή. What’s the difference?

αδερφή and αδελφή are the same word (sister), pronounced the same way: [aðerˈfi].

αδερφή is the more common modern spelling in everyday Greek.
αδελφή is a more traditional / conservative spelling (closer to the ancient form ἀδελφή), and you may see it in formal writing, religious texts, older literature, etc.

In speech, there is no difference; in modern casual writing, αδερφή is very frequent.

What tense and person is τραγουδάει, and what is the dictionary form of this verb?

τραγουδάει is:

Person / number: 3rd person singular (he/she/it sings)
Tense / mood: Present indicative
Verb: τραγουδάω / τραγουδώ (to sing)

Modern Greek verbs of this type often have two dictionary forms:

τραγουδάω (more “full” form)
τραγουδώ (shorter form; very common, especially in writing)

They both mean to sing. The 3rd person singular in everyday speech is typically τραγουδάει (or τραγουδά, see below).

Is there any difference between τραγουδάει and τραγουδά?

In modern spoken Greek, τραγουδάει and τραγουδά in this sentence are effectively the same in meaning: she sings.

τραγουδάει: slightly more colloquial in sound (everyday speech).
τραγουδά: a bit shorter and somewhat more common in writing, songs, headlines, etc.

Both are correct 3rd person singular present forms of τραγουδάω / τραγουδώ. You can say:

Η αδερφή μου τραγουδάει πολύ καλά.
Η αδερφή μου τραγουδά πολύ καλά.

with no real difference in meaning.

Why is it πολύ καλά and not πολύ καλό or something else?

Here:

πολύ = very (adverb of degree)
καλά = well (adverb)

So πολύ καλά means very well.

καλά is originally the neuter plural of the adjective καλός, but very often it functions as an adverb meaning well, just like κακό → κακά (badly), etc.

You would not use πολύ καλό here, because πολύ καλό is an adjective phrase (very good as a thing), not an adverb:

Τραγουδάει πολύ καλά. = She sings very well. (how she sings)
Το τραγούδι είναι πολύ καλό. = The song is very good. (what the song is like)

Could I change the word order and say Τραγουδάει πολύ καλά η αδερφή μου? Does that sound natural?

Yes, that is perfectly natural in Greek.

Both sentences are correct:

Η αδερφή μου τραγουδάει πολύ καλά.
Τραγουδάει πολύ καλά η αδερφή μου.

Greek word order is relatively flexible. Changing the word order can slightly shift emphasis:

Η αδερφή μου τραγουδάει πολύ καλά.
– Neutral: My sister sings very well.
Τραγουδάει πολύ καλά η αδερφή μου.
– Puts a bit more emphasis on τραγουδάει πολύ καλά (It’s very well that my sister sings / The one who sings very well is my sister — depending on context and intonation).

But in everyday conversation they both just sound natural.

How does θέλει να γίνει work? Why do we need να and a different verb form?

θέλει να γίνει is a very typical Greek structure:

θέλει = she wants (3rd person singular of θέλω, to want)
να = particle that introduces the subjunctive
γίνει = subjunctive form of γίνομαι (to become)

Greek often uses [verb] + να + subjunctive where English uses an infinitive:

θέλω να φάω = I want to eat
θέλει να γίνει μουσικός = She wants to become a musician

There is no infinitive in modern Greek, so structures like θέλω να…, πρέπει να…, μπορώ να… are extremely common.

What exactly is the form γίνει? Is it future, subjunctive, or something else?

γίνει is:

Verb: γίνομαι (to become / to be(come))
Tense / aspect: Aorist
Mood: Subjunctive
Person / number: 3rd person singular

So γίνει is the aorist subjunctive 3rd singular.

In θέλει να γίνει μουσικός, it expresses a change of state in the future (becoming a musician at some point), not a repeated or ongoing process. That’s why the aorist subjunctive (single event) is used, not a present subjunctive like γίνεται.

Why isn’t it γίνεται instead of γίνει? What’s the difference?

γίνει and γίνεται come from the same verb γίνομαι, but they differ in aspect and meaning:

να γίνει: aorist subjunctive → focuses on a single change or result: to become (once)
να γίνεται: present subjunctive → focuses on an *ongoing / repeated process: to be becoming / to keep becoming

In θέλει να γίνει μουσικός, the idea is “she wants to become (at some point) a musician”, a single life goal / change of state. So γίνει is the natural choice.

θέλει να γίνεται μουσικός would sound odd here; it would suggest a repeated or continuous becoming, which doesn’t fit the usual idea of choosing a profession.

Why is there no article before μουσικός? In English we say a musician.

In Greek, when talking about someone’s profession, role, or status using είμαι / γίνομαι and similar verbs, you usually omit the article:

Είμαι γιατρός. = I am a doctor.
Θα γίνω δάσκαλος. = I will become a teacher.
Είναι φοιτήτρια. = She is a student.
Θέλει να γίνει μουσικός. = She wants to become a musician.

So μουσικός here is a predicate noun describing what she will be, and it is indefinite (a musician). Greek normally does not use ένας/μία or ο/η in this pattern.

Does μουσικός show gender? How do we know it’s a female musician?

μουσικός is a common-gender noun: the same form is used for both men and women.

ο μουσικός = the (male) musician
η μουσικός = the (female) musician

In your sentence there is no article in front of μουσικός, so we determine the gender from context: Η αδερφή μου clearly indicates we are talking about a woman. So it means she wants to become a (female) musician.

If you added an article, it would show gender:

Θέλει να γίνει η μουσικός της ορχήστρας. = She wants to become the musician of the orchestra.
Θέλει να γίνει ο μουσικός της ορχήστρας. = could refer to a man.

Could we say θέλει να είναι μουσικός instead of θέλει να γίνει μουσικός? What’s the difference?

Both are grammatically correct, but they don’t mean exactly the same:

θέλει να γίνει μουσικός
- Literally: she wants to become a musician
- Focus on the change of state (from non-musician to musician).
- Implies a future goal: that’s what she wants to achieve in life.
θέλει να είναι μουσικός
- Literally: she wants to be a musician
- Focus on the state of being a musician, not on the process of becoming one.
- Could describe an ongoing wish: she wants her present status to be “musician” (for example, she doesn’t want to change careers).

In the context of life plans, θέλει να γίνει μουσικός is the most natural way to express she wants to become a musician.