Learn Hardcore Polish: Latem dzieci biegają boso po trawie obok tarasu i patrzą na górski krajobraz. - In the summer the children run barefoot on the grass next to the terrace and look at the mountain landscape.

You can say w lecie (“in the summer”) with w + locative, but Latem is shorter and very natural.
*w lato is not correct here; time phrases with seasons use either the instrumental alone (Latem) or w + locative (w lecie).

How does dzieci work grammatically? Why is it biegają and patrzą, not singular forms?

Dzieci means children and is grammatically plural (even though the singular is dziecko).

singular: dziecko – child
plural: dzieci – children

Verbs must agree with dzieci in the 3rd person plural:

Dzieci biegają. – The children run.
Dzieci patrzą. – The children look.

So you must use plural verb endings (-ą here):
*Dzieci biega or *Dzieci patrzy are ungrammatical in this context.

What does the ending -ają in biegają tell me?

Biegają is:

3rd person plural
present tense
imperfective aspect
from the verb biegać (to run around / to run, habitually)

The -ają ending is typical for 3rd person plural present tense of many -ać verbs:

biegać → biegają – they run
pływać → pływają – they swim
mieszkać → mieszkają – they live (somewhere)

Here Latem dzieci biegają… means “In summer, the children (usually / regularly) run…” – a repeated or typical action.

Why is it boso and not bose? What kind of word is boso?

Boso is an adverb, meaning “barefoot” / “barefooted.”

bosy – adjective, masculine (a barefoot boy: bosy chłopiec)
bosa – adjective, feminine
bose – adjective, neuter or non-masculine plural
boso – adverb (how they move? barefoot)

In Polish, when you describe how someone is doing an action, you use an adverb, not an adjective:

Dzieci biegają boso. – The children run barefoot.

Using *biegają bose would sound like you’re saying “they run being barefoot ones” – ungrammatical in this structure.

Why is it po trawie and not po trawa? Which case is used after po here?

After po with the meaning “on/along a surface; over; around”, Polish uses the locative case.

trawa – nominative (grass)
trawie – locative

So:

po trawie – (running) on/over the grass
po ulicy – along the street
po plaży – along the beach

In this sentence, po trawie is “(running) on the grass / across the grass,” so trawie must be in locative.

What’s the difference between po trawie and na trawie?

Both involve grass, but they suggest slightly different things:

na trawie – on the grass (location, being on top of it)
- Dzieci siedzą na trawie. – The children are sitting on the grass.
po trawie – on/over/around the grass (movement across the surface)
- Dzieci biegają po trawie. – The children run around on the grass.

In your sentence, biegają po trawie emphasizes movement across the grass, not just being located on it.

Why is it obok tarasu and not obok taras?

The preposition obok (“next to / beside”) requires the genitive case.

taras – nominative (terrace)
tarasu – genitive

So you must say:

obok tarasu – next to the terrace
obok domu – next to the house
obok szkoły – next to the school

Using *obok taras would be wrong because obok never takes nominative; it always needs genitive.

Why is it patrzą na górski krajobraz and not some other case? What does na do here?

The verb patrzeć (na) means “to look at” and always takes na + accusative:

patrzeć na coś – to look at something

Here:

krajobraz – nominative = krajobraz (masc. inanimate)
accusative (masc. inanimate) is identical to nominative → krajobraz

So:

patrzą na górski krajobraz – they are looking at the mountain landscape

Other verbs with na + accusative:

czekać na autobus – to wait for the bus
liczyć na pomoc – to count on help

How can I tell that górski krajobraz is in the accusative? It looks like nominative.

For masculine inanimate nouns, the accusative form is the same as the nominative.

nominative: krajobraz górski / górski krajobraz
accusative: krajobraz górski / górski krajobraz

So the form doesn’t change; you recognize the case from the preposition and the verb:

patrzeć na + accusative → so na górski krajobraz must be accusative.

With a feminine noun, you would see a change:

góra (nom.) → górę (acc.)
oglądam górę. – I’m looking at the mountain.

Is there any difference between górski krajobraz and krajobraz górski?

Both are grammatically correct and mean roughly “mountain landscape.”

górski krajobraz – a bit more neutral and common order: adjective + noun
krajobraz górski – can sound slightly more descriptive/emphatic, but in many contexts they are interchangeable.

In everyday speech, people tend to put adjectives before the noun: górski krajobraz.

Could I leave out dzieci since the verbs are in the plural? For example: Latem biegają boso…?

Grammatically, yes: Polish often drops the subject when it’s clear from context, because verb endings show person and number.

Biegają boso po trawie. – (They) run barefoot on the grass.

However:

At the beginning of a story or sentence, saying Dzieci makes it clear who is doing the action.
Without previous context, Latem biegają boso… might raise the question “Who?”

So it’s possible to omit dzieci, but in this standalone sentence including it is natural and clearer.

How is rz pronounced in patrzą and krajobraz? And what about the ą at the end of patrzą?

rz is usually pronounced like the voiced ż / sz-type sound:

rz ≈ ż (similar to the s in “measure”; IPA /ʐ/ or /ʒ/ depending on accent)

So:

patrzą – roughly PA-chʐon (the trz is like t
- sz/ż; many learners hear it as “tsh”)
krajobraz – KRA-yo-braz (with rz ≈ ż)

The final ą in patrzą:

At the end of a word, ą is nasal and often sounds like “on / om”:
- patrzą ≈ PAT-shon

Importantly, the ending -ą here is the 3rd person plural marker for many verbs:

biegają – they run
patrzą – they look
czytają – they read

What’s the difference between patrzą na krajobraz, oglądają krajobraz, and widzą krajobraz?

All involve vision, but the nuance differs:

patrzeć na coś – to look at something (direction of your eyes; neutral about whether you really notice details)
- Dzieci patrzą na górski krajobraz. – They’re looking at the mountain landscape.
oglądać coś – to watch / to look at carefully / to view (more active, longer, or more deliberate)
- Dzieci oglądają górski krajobraz. – They’re admiring/viewing the mountain landscape.
widzieć coś – to see something (perception itself; whether it enters your field of vision)
- Dzieci widzą górski krajobraz. – They can see the mountain landscape.

In your sentence, patrzą na górski krajobraz emphasizes that they direct their gaze toward the landscape.