Muszę wynieść worek do śmietnika.

In Polish, the verb ending usually already tells you who the subject is.
Muszę means I must / I have to, so ja is not necessary.

• muszę = I must
• musisz = you must
• musi = he/she/it must

You can add ja for emphasis, for example:

• Ja muszę wynieść worek. = I have to take out the bag.
  (implying: not someone else)

So the version without ja is the most neutral and natural one.

Wynieść means to carry/take something out, usually from inside to outside.

In this sentence, it suggests taking the bag out of the room, apartment, or house and bringing it to the trash bin/dumpster.

The prefix wy- often gives the idea of out/outward, so:

• nieść = to carry
• wynieść = to carry out / take out

So wynieść worek is very natural for take out the bag.

This is a question of aspect, which is very important in Polish.

• wynieść = perfective
• wynosić = imperfective

Here, wynieść is used because the speaker means one complete action: taking the bag out once, as a finished task.

• Muszę wynieść worek. = I have to take the bag out.
  (one completed action)

Compare:

• Muszę wynosić śmieci codziennie. = I have to take out the trash every day.
  (habit/repeated action)

So in your sentence, the perfective verb fits because it refers to a single concrete thing that needs to get done.

Because worek is a masculine inanimate noun, and in the accusative singular, masculine inanimate nouns usually look the same as the nominative singular.

So:

• Nominative: worek = bag/sack
• Accusative: worek = bag/sack

Since worek is the direct object of wynieść, it is in the accusative:

• Muszę wynieść worek.

If it were a masculine animate noun, the accusative would usually look like the genitive instead.

Because the preposition do requires the genitive case.

So:

• śmietnik = trash bin / dumpster / garbage area
• do śmietnika = to the trash bin / into the dumpster

This is a very common pattern in Polish:

• do domu = to the house / home
• do sklepu = to the shop
• do pokoju = to the room

So after do, you should expect the noun to change form.

Śmietnik usually refers to a place or container for rubbish, often something like:

• a dumpster
• a large outdoor trash bin
• a garbage area

Depending on context, it can sometimes be translated as trash can, but often it feels a bit bigger or more like the place where garbage is taken.

Useful contrast:

• kosz / kosz na śmieci = trash can / wastebasket, often indoors
• śmietnik = dumpster, garbage bin, or trash area
• śmieci = trash / garbage

So do śmietnika often sounds like taking the bag out to the main garbage container, not just dropping it into a small indoor bin.

You probably could, but it changes the nuance.

• wynieść worek do śmietnika = take/carry the bag out to the dumpster
• wyrzucić worek do śmietnika = throw the bag away into the dumpster

So:

• wynieść focuses on taking it out
• wyrzucić focuses on throwing it away/discarding it

In everyday speech, if you're talking about household trash, wynieść worek is very natural because the task is commonly understood as taking the trash bag out.

Because the idea of out is built into the verb wynieść through the prefix wy-.

English often uses:

• take out
• carry out

Polish often expresses that meaning with a prefixed verb:

• nieść = carry
• wynieść = carry out
• wnieść = carry in
• zanieść = take/carry somewhere
• przynieść = bring

So instead of a separate word like out, Polish often uses a verb prefix.

Not exactly.

• worek = sack, bag, usually a softer or more utilitarian bag, such as a trash bag, sack of potatoes, etc.
• torba = bag, often a shopping bag, handbag, tote bag, or general carry bag

For a trash bag, worek is the natural choice:

• worek na śmieci = trash bag

So in this sentence, worek strongly suggests a garbage bag or sack rather than an ordinary bag.

Yes. Polish word order is more flexible than English, although not all versions sound equally neutral.

The basic neutral order here is:

• Muszę wynieść worek do śmietnika.

But you could also hear:

• Worek muszę wynieść do śmietnika.
  (emphasizes the bag)
• Do śmietnika muszę wynieść worek.
  (emphasizes to the trash bin)

Because Polish uses endings and case forms, word order is often used for focus and emphasis, not just grammar.

A rough pronunciation guide is:

MOO-sheh vih-NYESHCH VOH-rek doh shmyet-NEE-kah

A few helpful notes:

• sz sounds like English sh
• ś is a softer sound, somewhat like a softer sh
• ę in muszę is often pronounced approximately like e or en/em depending on context, but learners can safely start with something close to mushe
• nieść has a soft ń sound, so wynieść is not pronounced like plain nie-sc
• stress in Polish is usually on the second-to-last syllable:
  • MU-szę
  • wy-NIEŚĆ
  • WO-rek
  • śmiet-NI-ka

A natural rhythm would be:

MUSZĘ wyNIEŚĆ WOREK do śmietNIKA.