Learn Hardcore Polish: W tym garnku gotuje zupę, a na patelni smaży warzywa. - In this pot she cooks soup, and in the frying pan she fries vegetables.

gotuje and smaży are 3rd person singular present tense forms.
So the sentence means he is cooking / she is cooking / (someone) is cooking, depending on context.

To make it explicit, you could say:

On w tym garnku gotuje zupę, a na patelni smaży warzywa. – He…
Ona w tym garnku gotuje zupę, a na patelni smaży warzywa. – She…

Context (earlier sentences, situation) usually tells you who it is, so pronouns are dropped unless needed for clarity or emphasis.

What’s the difference between gotuje and gotuję?

Both come from the verb gotować (to cook), but they have different persons:

gotuję – I cook / I am cooking (1st person singular)
gotuje – he/she/it cooks / is cooking (3rd person singular)

In the sentence:

gotuje zupę = he/she is cooking soup (or someone is cooking soup).

If you were talking about yourself, you’d say:

W tym garnku gotuję zupę. – In this pot I am cooking soup.

Why is it zupę and not zupa?

Zupa is a feminine noun:

nominative (dictionary form): zupa
accusative (direct object): zupę

Here, zupę is the direct object of the verb gotuje (what is (s)he cooking?), so Polish uses the accusative case:

(Ktoś) gotuje zupę. – Someone is cooking soup.

Use zupa when it is the subject:

Zupa jest gorąca. – The soup is hot.

Why is it warzywa and not something like warzywy?

Warzywo (vegetable) is a neuter noun.

Its plural nominative and accusative form is warzywa:

singular: warzywo
plural: warzywa

In the sentence, warzywa are the direct object of smaży (what is (s)he frying?), so they are in the accusative plural. For neuter nouns, nominative plural and accusative plural look the same:

Warzywa są zdrowe. – Vegetables are healthy. (subject)
Smaży warzywa. – (He/She) is frying vegetables. (object)

Why does garnek change to garnku in w tym garnku?

Garnek (pot) is a masculine noun. Garnku is its locative case form.

After the preposition w meaning in (location, no movement), Polish requires the locative case:

nominative: garnek – pot (dictionary form)
locative singular: garnku

So:

w garnku – in the pot (location)
w tym garnku – in this pot

If you are talking about putting something into the pot (movement), you typically use do + genitive:

Wkładam warzywa do garnka. – I am putting vegetables into the pot.

Why is it w tym garnku and not w ten garnek?

Two things are going on:

Meaning (location vs movement)
- w + locative = in (where something is)
- w + accusative (much less common here) would suggest movement into something, and even then speakers usually prefer do garnka.

In your sentence, we talk about where the soup is being cooked (location), so we use locative:

w tym garnku – in this pot (no movement)

Form of the demonstrative “this”
- masculine nominative: ten garnek – this pot (as subject)
- masculine locative: w tym garnku – in this pot

So tym is simply the locative form of ten.

Why is it na patelni and not na patelnię?

The preposition na also behaves like w:

na + locative = on / on top of (location, no movement)
na + accusative = onto (movement, direction)

In na patelni smaży warzywa, we are describing where the vegetables are being fried (on the pan), not movement onto it, so we use locative:

patelnia – pan (nominative)
na patelni – on the pan (locative, location)

If you were talking about putting something onto the pan (movement), you could say:

Kładzie warzywa na patelnię. – He/She is putting vegetables onto the pan. (accusative)

Why is it w tym garnku but na patelni? Why different prepositions: w and na?

Polish distinguishes inside vs on the surface more strongly than English:

w
- locative = in (inside something, container/space)
  - w tym garnku – in this pot (inside the pot)
na
- locative = on (on a surface)
  - na patelni – on the pan (on the surface of the pan)

English often says “in the pan”, but physically the food lies on the flat surface, so Polish uses na patelni.

Can I change the word order? For example: Gotuje zupę w tym garnku, a smaży warzywa na patelni?

Yes. Polish word order is quite flexible, especially for adverbial phrases like w tym garnku and na patelni.

All of these are correct and natural:

W tym garnku gotuje zupę, a na patelni smaży warzywa.
Gotuje zupę w tym garnku, a smaży warzywa na patelni.
Zupę gotuje w tym garnku, a warzywa smaży na patelni.

The differences are mostly about emphasis and rhythm, not grammar. Starting with w tym garnku highlights the place first; putting it at the end can sound more neutral or like additional information.

Do I have to use tym and patelni? Can I just say w garnku gotuje zupę, a na patelni smaży warzywa?

You can omit tym and it will still be correct:

W garnku gotuje zupę, a na patelni smaży warzywa.

Polish has no articles (a/the), so:

w garnku can mean in a pot or in the pot depending on context.
w tym garnku = in this pot (more specific, demonstrative)
w tym garnku can also sometimes behave like stressing in that/this particular pot.

So:

With tym → more specific, like English this/that.
Without tym → more neutral, like a pot / the pot from context.

Why is the conjunction a used instead of i? Aren’t they both “and”?

Both a and i can be translated as and, but they have different nuances:

i – simple addition, links similar actions, things, or ideas (often like “and also”).
a – contrasts or distinguishes two parts; often like “and meanwhile / whereas / while”.

In your sentence:

W tym garnku gotuje zupę, a na patelni smaży warzywa.

a suggests a contrast between locations and actions:

In this place, one thing is happening, whereas in that place, another thing is happening.

You could say i here, but a sounds more natural because of the contrast: pot vs pan, boiling vs frying.

Is gotuje / smaży more like “cooks” or “is cooking”? Does Polish have a separate “-ing” form?

Polish does not have a special continuous form like English is cooking.

The present tense often covers both:

gotuje = he/she cooks or he/she is cooking
smaży = he/she fries or he/she is frying

Context decides whether it is:

habitual:
Codziennie gotuje zupę. – He/She cooks soup every day.
right now / ongoing:
Teraz w tym garnku gotuje zupę. – Right now he/she is cooking soup in this pot.

If you want to stress “right now”, you can add words like teraz (now), właśnie (just now), w tej chwili (at the moment).

What would the sentence look like if the subject was they instead of he/she?

For they, the verb must be 3rd person plural:

Oni (they, group with at least one man) / one (they, all-female or non‑male things)

Then:

W tym garnku gotują zupę, a na patelni smażą warzywa.
– They are cooking soup in this pot, and frying vegetables in the pan.

Changes:

gotuje → gotują (3rd person singular → plural)
smaży → smażą (3rd person singular → plural)

The rest of the sentence stays the same, because zupę, warzywa, w tym garnku, na patelni do not depend on the subject’s number.