Chcę wziąć udział w kursie.

Because Polish often leaves out subject pronouns when they are obvious from the verb form.

Chcę already means I want, so ja is not necessary.

You would add ja only for emphasis or contrast, for example:

• Ja chcę wziąć udział w kursie, a on nie.
  I want to take part in the course, but he doesn’t.

So the sentence without ja is the normal, neutral version.

In Polish, the infinitive does not use a separate word like English to.

So:

• chcę wziąć = I want to take
• chcę iść = I want to go
• chcę czytać = I want to read

The infinitive itself is the basic dictionary form of the verb, and here that form is wziąć.

Wziąć udział is a fixed expression meaning to take part or to participate.

It is best learned as a whole chunk:

• wziąć udział w czymś = to take part in something

Literally, udział can mean share or part, but in this expression the whole phrase functions like one idea. So it is better not to translate it word by word every time.

This is about aspect, which is very important in Polish verbs.

• brać udział = imperfective
• wziąć udział = perfective

A simple way to think about it:

• brać udział focuses on the activity as ongoing, repeated, or general
• wziąć udział focuses on taking part as a single complete event

So:

• Chcę wziąć udział w kursie suggests wanting to participate in this course as a whole event
• Chcę brać udział w kursie is also possible, but it can sound more like ongoing participation

Both are grammatical, but wziąć udział is very common when referring to one specific event or course.

Because the expression wziąć udział requires w:

• wziąć udział w czymś
• brać udział w czymś

So after this phrase, you use w plus the correct case.

That is why you get:

• w kursie = in the course

However, with other verbs or expressions, na kursie can also be correct:

• Jestem na kursie. = I’m on/at a course.
• Byłem na kursie. = I was on/at a course.

So the choice depends on the whole expression, not only on the noun kurs.

Because after w in this meaning, Polish uses the locative case.

So:

• basic form: kurs
• locative singular: kursie

That is why the sentence has:

• w kursie

This is a normal pattern in Polish. Compare:

• w domu = in the house
• w sklepie = in the shop
• w teatrze = in the theatre
• w kursie = in the course

So -ie here is a case ending.

Yes. In this sentence, udział is part of the fixed phrase wziąć udział.

Grammatically, it works like the object of the verb wziąć, but for learners the most useful thing is to memorize the whole expression:

• wziąć udział w czymś
• brać udział w czymś

That will help you much more than trying to build it from separate words each time.

Yes, absolutely.

Uczestniczyć w kursie also means to participate in a course.

So these are both natural:

• Chcę wziąć udział w kursie.
• Chcę uczestniczyć w kursie.

A small difference in feel:

• uczestniczyć is a straightforward verb meaning to participate
• wziąć udział is a very common expression meaning to take part

Both are correct and useful. Uczestniczyć can sound a little more formal in some contexts.

Yes, Polish word order is more flexible than English word order.

The neutral, most natural version here is:

• Chcę wziąć udział w kursie.

But other orders are possible for emphasis, for example:

• W kursie chcę wziąć udział.

That might emphasize in the course.

Still, for learners, the original order is the safest and most natural choice.

These two words are tricky for English speakers.

A rough guide:

• chcę: the ch is like a hard Polish h sound, not English ch
• wziąć: the w is pronounced like English v, not like English w

A more technical guide:

• chcę ≈ /xt͡sɛ̃/
• wziąć ≈ /vʑɔɲt͡ɕ/

A few practical tips:

• ę is a nasal vowel
• Polish w sounds like English v
• clusters like chc and wz are normal in Polish, even though they feel difficult at first

So if you pronounce them slowly and clearly at first, that is completely fine.