Az ifjúsági program holnap kezdődik.

A word-by-word breakdown is:

• Az = the
• ifjúsági = youth, youth-related, for young people
• program = program
• holnap = tomorrow
• kezdődik = begins, starts

So the whole sentence means something like:

• The youth program starts tomorrow.
• The youth program begins tomorrow.

Hungarian has two forms of the definite article:

• a
• az

They both mean the.

The choice depends on the sound that follows:

• a before a consonant
• az before a vowel

Since ifjúsági begins with a vowel sound, Hungarian uses az:

• az ifjúsági program

Compare:

• a program = the program
• az iskola = the school

Ifjúsági is an adjective meaning youth, for young people, or youth-related.

It comes from the noun ifjúság, which means youth or young people, plus the adjective-forming ending -i.

So:

• ifjúság = youth
• ifjúsági = youth / youth-related

In this sentence, ifjúsági program means a youth program, not a program that is itself young.

This is a very common learner question.

• kezd usually means starts something / begins something
  It is typically used transitively, with a direct object.
• kezdődik means begins / starts on its own
  It is intransitive.

So:

• Kezdem a munkát. = I’m starting the work.
• A munka kezdődik. = The work is starting.

In your sentence, the program is not starting something else; the program itself is beginning. So Hungarian uses:

• Az ifjúsági program holnap kezdődik.

This is very natural Hungarian.

Because Hungarian very often uses the present tense for future events when the time is already clear from the context.

Here, holnap already tells us the event is in the future:

• holnap = tomorrow

So Hungarian does not need a separate future form here. This is similar to English sentences like:

• The program starts tomorrow.
• We leave next week.

Even though the verb looks present, the meaning is future because of the time word.

Yes, but Hungarian word order is more flexible than English word order.

The neutral version is:

• Az ifjúsági program holnap kezdődik.

This is a straightforward statement: The youth program starts tomorrow.

You could also say:

• Holnap kezdődik az ifjúsági program.

This puts more focus on holnap or makes the sentence feel a bit more like It’s tomorrow that the youth program starts.

So the meaning stays very close, but the emphasis can change.

Because holnap is already an adverb meaning tomorrow.

Hungarian often uses case endings on nouns, but some time expressions are standalone adverbs and do not need an extra ending.

For example:

• ma = today
• holnap = tomorrow
• tegnap = yesterday

So in this sentence, holnap is simply the adverb tomorrow.

A rough pronunciation guide is:

• ifjúsági ≈ eef-yoo-shaa-ghee

A few helpful points:

• Hungarian stress is usually on the first syllable: IFjúsági
• ú is a long vowel, like a longer oo
• s in Hungarian sounds like English sh
• g is always a hard g, as in go

So the middle part sá sounds like shaa.

A rough guide is:

• kezdődik ≈ KEZ-duh-dik

More precisely:

• e is like the e in bed
• ő is a rounded vowel that English does not really have; learners often approximate it as something between uh and er, but with rounded lips
• stress is on the first syllable: KEZdődik

Also note:

• Hungarian spelling is very regular, so once you learn the sound system, pronunciation becomes much easier.

Yes. Program is a common international word, and Hungarian uses it too.

Its meaning depends on context, just like in English:

• a schedule or event program
• a plan
• a computer program

In ifjúsági program, it means a program/event/organized activity for young people.

Yes. In natural English, several translations work:

• The youth program starts tomorrow.
• The youth program begins tomorrow.
• The youth program is starting tomorrow.

The Hungarian sentence is simple and neutral, so all of these can fit depending on context.