W sobotę moja siostra wybierze spódnicę i sandały, a ja zostanę przy dżinsach i kapeluszu.

Because after w when talking about on Saturday, Polish uses the accusative form here.

• sobota = nominative, the basic dictionary form
• sobotę = accusative

So:

• w sobotę = on Saturday

This is very common with days of the week:

• w poniedziałek = on Monday
• we wtorek = on Tuesday
• w środę = on Wednesday
• w sobotę = on Saturday
• w niedzielę = on Sunday

So w sobotę is just the normal idiomatic way to say on Saturday.

Because moja siostra is the subject of the sentence, so it stays in the nominative case.

Here, moja siostra wybierze... means my sister will choose...

• moja siostra = nominative, used for the subject
• moją siostrę = accusative, used when my sister is the object

Compare:

• Moja siostra wybierze spódnicę. = My sister will choose a skirt.
• Widzę moją siostrę. = I see my sister.

So in your sentence, moja siostra is correct because she is the one doing the choosing.

Wybierze is future tense, third person singular: he/she/it will choose.

It comes from the verb pair:

• wybierać = to be choosing / to choose repeatedly or imperfectively
• wybrać = to choose, to pick (perfective)

wybierze is the future form of the perfective verb. In Polish, perfective verbs usually form the future with a single word:

• wybierze = she will choose

So:

• moja siostra wybierze spódnicę i sandały = my sister will choose a skirt and sandals

Because they are the direct objects of wybierze.

The verb wybrać / wybierać takes the accusative case for what is being chosen.

• spódnica → spódnicę
  feminine singular accusative
• sandały → sandały
  masculine plural accusative, which here looks the same as the nominative plural

So:

• wybierze spódnicę = will choose a skirt
• wybierze sandały = will choose sandals

This is very normal in Polish: direct objects often go into the accusative.

Because different noun types change differently in the accusative.

spódnica

This is a feminine singular noun:

• nominative: spódnica
• accusative: spódnicę

So it changes form.

sandały

This is a plural noun. In this case, the accusative plural is the same as the nominative plural:

• nominative plural: sandały
• accusative plural: sandały

So it looks unchanged, even though it is functioning as an object.

This is something learners see a lot in Polish: sometimes the case changes are visible, and sometimes the form happens to stay the same.

A often links two clauses with a contrast or comparison, softer than but.

In this sentence:

• moja siostra wybierze...
• a ja zostanę...

the meaning is something like:

• my sister will choose..., and as for me / while I will...
• or my sister will choose..., but I’ll stick with...

Difference:

• i = simple and
• a = and/but/while with contrast
• ale = stronger but

So a ja is very natural here because the speaker is contrasting their choice with the sister’s choice.

Here zostanę przy means I’ll stick with or I’ll stay with.

Literally, zostać przy can mean to remain by/with, but in natural English here it means:

• I’ll stick with jeans and a hat
• I’ll stay with jeans and a hat

So the whole second part:

• a ja zostanę przy dżinsach i kapeluszu

means something like:

• and I’ll stick with jeans and a hat

This is a common Polish structure:

• zostać przy czymś = to stick with something

Because przy requires the locative case.

So after przy, the nouns must change:

• dżinsy → dżinsach
• kapelusz → kapeluszu

That is why the sentence says:

• przy dżinsach i kapeluszu

not:

• przy dżinsy i kapelusz

This is an important rule to remember:

• przy + locative

Examples:

• przy stole = by the table
• przy oknie = by the window
• zostać przy swoim zdaniu = stick to one’s opinion
• zostać przy dżinsach = stick with jeans

Usually, yes. In normal Polish, dżinsy is typically used as a plural noun, like jeans in English.

So you usually get forms such as:

• dżinsy = jeans
• dżinsów
• dżinsach
• dżinsami

That is why the sentence says przy dżinsach.

In practice, you should learn dżinsy as a plural clothing noun, similar to:

• spodnie = trousers/pants
• okulary = glasses

Because kapelusz has a locative form kapeluszu, and that is the standard form after przy.

• nominative: kapelusz
• locative: kapeluszu

So:

• przy kapeluszu would be the correct locative pattern if the noun stood alone
• in your sentence: przy dżinsach i kapeluszu

Some Polish nouns have locative endings that may not be obvious to learners, so it is best to learn them with the preposition:

• przy kapeluszu
• o kapeluszu

Both can refer to the future, but they are not the same.

wybierze

This is the future of a perfective verb. It focuses on the action as a completed choice:

• wybierze = will choose / will pick

będzie wybierać

This is the future of an imperfective verb. It focuses more on the process, repetition, or ongoing action:

• będzie wybierać = will be choosing

In your sentence, wybierze is more natural because the idea is that the sister will make a choice.

Yes, Polish word order is fairly flexible, but the original version is natural and clear:

• W sobotę moja siostra wybierze spódnicę i sandały, a ja zostanę przy dżinsach i kapeluszu.

You could move parts around for emphasis, for example:

• Moja siostra w sobotę wybierze spódnicę i sandały...
• Ja zostanę przy dżinsach i kapeluszu...
• Przy dżinsach i kapeluszu zostanę ja... (very marked/emphatic)

But the original word order sounds the most neutral and natural for everyday Polish.

A few parts may be difficult for English speakers:

• sobotę: the final ę is usually pronounced somewhat like e or a nasalized vowel depending on context
• siostra: si sounds like a soft sh/ś-type sound, not a hard English s
• wybierze: the rz here sounds like ż
• dżinsach: dż sounds like the j in jam
• kapeluszu: the sz sounds like English sh
• ł in Polish sounds like English w

A rough pronunciation guide:

• W sobotę ≈ f so-BO-te
• siostra ≈ SHO-stra
• wybierze ≈ vih-BYE-zhe
• dżinsach ≈ JEEN-sakh
• kapeluszu ≈ ka-pe-LOO-shu

These are only rough approximations, but they can help at the beginning.