Chcę nauczyć się mówić po polsku.

A quick breakdown:

• chcę = I want
  First person singular of chcieć.

• nauczyć się = to learn / to master / to get oneself to the point of knowing how
  This is a perfective infinitive.

• mówić = to speak
  An infinitive describing the skill being learned.

• po polsku = in Polish
  The usual way to say that something is done in the Polish language.

So the structure is: I want + to learn + to speak + in Polish.

Because Polish usually drops subject pronouns when they are already clear from the verb form.

• chcę already means I want
• so ja chcę is possible, but ja is usually only added for emphasis, contrast, or clarity

For example:

• Chcę nauczyć się mówić po polsku. = neutral
• Ja chcę nauczyć się mówić po polsku. = I want to learn to speak Polish (maybe not someone else)

Chcę is the 1st person singular form of chcieć: I want.

Some forms of chcieć are:

• chcę = I want
• chcesz = you want
• chce = he/she/it wants
• chcemy = we want
• chcecie = you plural want
• chcą = they want

So chce would mean he/she/it wants, not I want.

This is mainly about aspect.

• uczyć się is imperfective: it focuses on the process of learning
• nauczyć się is perfective: it focuses on reaching the result, meaning to learn successfully / to master

In this sentence, the speaker wants to reach a goal: to become able to speak Polish. That is why nauczyć się sounds natural.

Compare:

• Chcę uczyć się mówić po polsku.
  More like: I want to be learning how to speak Polish
  Focus on the process

• Chcę nauczyć się mówić po polsku.
  I want to learn how to speak Polish
  Focus on achieving the skill

In this sentence, się is part of the verb nauczyć się.

It is often called a reflexive particle, but here you should mostly learn the whole verb together:

• uczyć się = to learn
• nauczyć się = to learn/master

It does not literally mean oneself in a way that English would normally translate word-for-word.

So it is best to treat nauczyć się as one unit when learning the phrase.

Because Polish, like English, can chain verbs together when one verb describes another.

Here:

• nauczyć się = to learn
• mówić = to speak

So nauczyć się mówić means to learn to speak.

English uses the extra word to before speak, but Polish does not need a separate word there. The infinitive form itself already does that job.

You see the same pattern in other sentences:

• Chcę iść. = I want to go.
• Umiem pływać. = I can swim.
• Muszę pracować. = I have to work.

Because mówić is the normal verb for speaking a language.

• mówić po polsku = to speak Polish

It also makes sense aspectually: speaking a language is an ongoing skill, not a one-time completed action.

Compare with other verbs:

• powiedzieć = to say/tell once, as a completed act
  Not right here

• rozmawiać = to have a conversation / to chat
  Related, but narrower than speak a language

So if you are talking about language ability, mówić po polsku is the natural phrase.

Because in Polish, the usual way to say in a language is:

• po + language form

So:

• po polsku = in Polish
• po angielsku = in English
• po niemiecku = in German
• po francusku = in French

That means:

• mówić po polsku = to speak Polish

You do not normally say mówić polski for this meaning.

So this is a set pattern that is worth memorizing as a chunk: mówić po polsku

Because in Polish, names of languages and adjectives like polski, angielski, niemiecki are normally written with a lower-case letter.

So:

• po polsku
• język polski
• mówię po angielsku

This is different from English, where Polish and English are capitalized.

Yes, and it is a very useful alternative, but it means something slightly different in focus.

• Chcę nauczyć się polskiego.
  = I want to learn Polish.
  This is broader and can include grammar, vocabulary, reading, writing, etc.

• Chcę nauczyć się mówić po polsku.
  = I want to learn to speak Polish.
  This specifically highlights speaking ability.

So both are correct, but the second one is more specific.

Yes, Chcę się nauczyć mówić po polsku is also natural.

Both of these are acceptable:

• Chcę nauczyć się mówić po polsku.
• Chcę się nauczyć mówić po polsku.

The particle się is fairly mobile in Polish, although it usually does not go at the very beginning of a clause.

The difference here is small and mostly about rhythm or emphasis, not basic meaning.

Many learners will hear both patterns, so it is good to recognize both.