A tanfolyam áprilisban kezdődik.

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

So:

• a tanfolyam = the course

Hungarian has two forms of the definite article:

• a before consonant sounds
• az before vowel sounds

Since tanfolyam starts with t, it takes a.

Tanfolyam usually means a course, especially an organized course of study such as a language course, training course, or evening course.

It is not usually the same as a single lesson or class session.
So if the meaning shown is the course begins in April, that fits tanfolyam very well.

Hungarian often uses suffixes where English uses prepositions.

Here, -ban / -ben means in.

So:

• április = April
• áprilisban = in April

This is very typical in Hungarian:

• januárban = in January
• májusban = in May
• budapesten = in Budapest

So instead of a separate word like English in, Hungarian attaches the idea directly to the noun.

This is because of vowel harmony, an important Hungarian pattern.

Very simply:

• words with mostly back vowels usually take -ban
• words with mostly front vowels usually take -ben

Április contains back vowels overall in the form that takes this suffix, so the natural form is:

• áprilisban

You do not need to think too mechanically about every word at first; it is often best to learn the suffixed form together with the base word.

Kezdődik means begins / starts in an intransitive sense — the event itself begins.

So:

• A tanfolyam áprilisban kezdődik. = The course begins in April.

By contrast, kezd is often used when someone starts something:

• A tanár elkezdi a tanfolyamot. = The teacher starts the course.

A useful way to feel the difference is:

• kezdődik = it begins
• kezd = someone starts it

The verb form is present tense:

• kezdődik = begins / is beginning

But Hungarian often uses the present tense to talk about the future, especially for scheduled or definite future events.

So this sentence literally looks present, but in meaning it can naturally be:

• The course begins in April
• The course will begin in April

Both are good English translations depending on context.

Yes. That is completely natural.

Hungarian word order is flexible, and changing the order often changes the focus or emphasis, not the basic meaning.

Compare:

• A tanfolyam áprilisban kezdődik.
  Neutral: The course begins in April.

• Áprilisban kezdődik a tanfolyam.
  Puts more focus on when it begins: It’s in April that the course begins.

So both are correct, but they highlight different information.

In Hungarian, the names of months are normally not capitalized.

So Hungarian writes:

• január
• február
• április

This is different from English, where month names are capitalized.

So áprilisban is correct Hungarian spelling.

A rough pronunciation guide is:

• kez sounds a bit like kez
• dő has a long ő, a rounded vowel that does not exist in English
• dik sounds like deek but shorter and crisper

Very roughly: KEZ-dur-dik is not exact, but may help at first.

A few useful points:

• Stress in Hungarian is almost always on the first syllable: KEZ-dő-dik
• ő is a long vowel
• a in Hungarian is not like English ay; it is more open

Normally, no, not in this neutral sentence.

If you mean the course, Hungarian usually wants the definite article:

• A tanfolyam áprilisban kezdődik.

Leaving out the article would sound incomplete or unnatural in most ordinary contexts.

Hungarian articles do not always match English perfectly, but in this sentence the article is the normal choice.

It breaks down like this:

• A tanfolyam = the course
• áprilisban = in April
• kezdődik = begins

So the structure is:

• subject + time expression + verb

That is a very common Hungarian sentence pattern.