A tanfolyam júniusban kezdődik.

A is the Hungarian definite article, equivalent to the in English.

So:

• a tanfolyam = the course

Hungarian has two forms of the definite article:

• a before consonant sounds
• az before vowel sounds

Since tanfolyam begins with t, it takes a.

Tanfolyam usually means a course, especially a class, training course, or organized program of study.

It is a noun. In this sentence:

• a tanfolyam = the course

Depending on context, it could refer to a language course, a training course, a workshop series, and so on.

Hungarian often adds endings directly to nouns instead of using separate words like English prepositions.

Here:

• június = June
• júniusban = in June

The ending -ban / -ben means in.

Examples:

• házban = in the house
• iskolában = in the school
• júniusban = in June

So Hungarian says June-in rather than using a separate word like in.

Hungarian vowel harmony determines which version of a suffix is used.

The suffix has two forms:

• -ban
• -ben

Words with mostly back vowels usually take -ban.
Words with mostly front vowels usually take -ben.

Június takes -ban, so:

• júniusban

This is something learners gradually get used to by exposure. The important point here is that június requires the -ban form.

Kezdődik means begins / starts.

It comes from the verb kezdődik, which is an intransitive verb, meaning the subject itself begins:

• A tanfolyam kezdődik. = The course begins.

By contrast, kezd is usually used when someone begins something:

• A tanár kezdi az órát. = The teacher starts the lesson.

So:

• kezdődik = begins / starts
• kezd = starts something

In this sentence, the course itself is beginning, so kezdődik is the natural choice.

It is in the present tense, third person singular.

Breakdown:

• dictionary form: kezdődik
• subject: a tanfolyam = the course
• tense: present

Hungarian often uses the present tense to talk about scheduled future events, just like English can:

• The course begins in June.
• The train leaves tomorrow.

So even though the event may be in the future, the present tense is normal here.

Because Hungarian usually expresses meanings like in, on, to, from, and so on with suffixes attached to the noun.

English:

• in June

Hungarian:

• júniusban

So instead of a separate preposition, Hungarian adds an ending to the month name.

This is the neutral, natural word order:

• A tanfolyam = topic/subject
• júniusban = time expression
• kezdődik = verb

A very normal translation is:

• The course begins in June.

Hungarian word order is more flexible than English, but changes in order often change emphasis.

For example:

• Júniusban kezdődik a tanfolyam.
  = It’s in June that the course begins / emphasis on June

So the original sentence is the straightforward, neutral version.

A rough pronunciation guide:

• tanfolyam ≈ tawn-foh-yom
• júniusban ≈ yoo-nee-oosh-bawn
• kezdődik ≈ kez-doe-deek

A few helpful notes:

• j in Hungarian is pronounced like English y
• s is pronounced like English sh
• sz is pronounced like English s
• ő is a long front rounded vowel, somewhat like a longer version of the vowel in German schön or French deux

Pronunciation varies slightly by speaker, but those approximations are a good start.

Yes, often you can, depending on context.

• kezdődik = begins / starts
• indul = starts / sets off / begins

For a course, kezdődik is very common and direct.
Indul can also be used for courses, especially in announcements:

• A tanfolyam szeptemberben indul. = The course starts in September.

Both are possible, but kezdődik is a very clear and standard choice here.