Chociaż rytm był trudny, moja siostra szybko nauczyła się drugiej zwrotki.

Chociaż means although / even though. It introduces a subordinate clause that gives a contrast:

• Chociaż rytm był trudny = Although the rhythm was difficult
• moja siostra szybko nauczyła się drugiej zwrotki = my sister quickly learned the second verse

In Polish, a clause introduced by chociaż is normally separated by a comma from the main clause, so the comma is required here.

Because rytm is a masculine singular noun.

So both parts match that noun:

• był = was for masculine singular in the past tense
• trudny = difficult, masculine singular form

Compare:

• rytm był trudny = the rhythm was difficult
• melodia była trudna = the melody was difficult

The gender of the noun changes the form of the past tense and the adjective.

Because siostra is a feminine noun, so the possessive pronoun must agree with it.

• moja siostra = my sister
• mój brat = my brother
• moje dziecko = my child

In Polish, words like mój / moja / moje change to match the gender and case of the noun they describe.

Nauczyła się is the past tense, feminine singular form of nauczyć się, which means to learn / to master.

It is feminine singular because the subject is moja siostra.

Breakdown:

• nauczyć się = to learn
• nauczyła się = she learned

The ending -a in the past tense often shows a feminine subject:

• nauczył się = he learned
• nauczyła się = she learned

In this sentence, się is part of the verb nauczyć się. You should learn it as a whole expression.

• nauczyć can mean to teach
• nauczyć się means to learn

So:

• Nauczyłam siostrę piosenki = I taught my sister a song
• Nauczyłam się piosenki = I learned the song

Here, się changes the meaning from teach to learn.

Because nauczyć się usually takes the genitive case.

The basic form is:

• druga zwrotka = nominative, the second verse

But after nauczyć się, it becomes genitive:

• drugiej zwrotki

So:

• nauczyć się czego? = to learn what?
• drugiej zwrotki = the second verse

Form changes:

• druga → drugiej
• zwrotka → zwrotki

Zwrotka usually means a verse or stanza, especially in a song or poem.

So druga zwrotka is:

• the second verse
• the second stanza

In song-related contexts, zwrotka is very natural. If you are thinking of a single line of poetry, Polish often uses wers instead.

Szybko means quickly, and Polish word order is fairly flexible. Putting the adverb before the verb is very natural:

• moja siostra szybko nauczyła się drugiej zwrotki

It could also appear in other places, for example:

• Moja siostra nauczyła się szybko drugiej zwrotki

That is also understandable, but the original version sounds smoother and more neutral.

Yes. Polish word order is more flexible than English, because endings show grammatical relationships.

For example, you could also say:

• Moja siostra szybko nauczyła się drugiej zwrotki, chociaż rytm był trudny.

This still means the same thing, but the emphasis shifts a little. The original sentence starts with the contrast:

• Although the rhythm was difficult...

That makes the difficulty feel like the background before the main point.

Because nauczyła się focuses on the result: she successfully learned it.

• uczyła się = she was learning / she studied
• nauczyła się = she learned it, she managed to master it

So in this sentence, the idea is not just that she spent time learning, but that she actually succeeded in learning the second verse. That is why nauczyła się fits better.