Learn Hardcore Czech: Centrum města je dnes večer plné turistů a lidí a přijde tam také náš učitel. - The city center is full of tourists and people this evening, and our teacher will also come there.

The second noun (města) in genitive answers the question “center of what?”.
You cannot just put two nouns next to each other in nominative (centrum město would be wrong).

You could also say městské centrum (urban/city center), but that slightly changes the nuance: it sounds more like a type of center, not specifically “the (geographic) center of this town.”

Why is města in the genitive case here?

Because Czech often uses the genitive to show possession or a close relationship between two nouns:

centrum města – the city’s center / the center of the city
okraj lesa – the edge of the forest
střed Evropy – the center of Europe

In all of these, the second noun is in genitive, answering “of what?”
So města is genitive singular of město.

Why is it plné turistů a lidí and not plný turistů a lidí?

The adjective must agree with centrum, which is:

centrum – neuter, singular, nominative

The nominative neuter singular form of plný is plné. So:

centrum je plné (neuter singular)
park je plný (masculine singular)
ulice je plná (feminine singular)
ulice jsou plné (feminine plural)

So plné matches centrum in gender (neuter) and number (singular).

Why are turistů and lidí in the genitive case after plné?

The adjective plný / plná / plné (full) typically takes the genitive to express “full of something”:

plný lidí – full of people
plná vody – full of water
plné turistů a lidí – full of tourists and people

So both turistů (genitive plural) and lidí (genitive plural) are governed by plné.

In other words, you can think:
plné (čeho?) turistů a lidí – full (of what?) of tourists and people.

What’s the difference between lidé and lidí?

Both come from the word člověk (person), whose plural is irregular:

lidé – nominative plural (subject form)
- Lidé zpívají. – People are singing.
lidí – genitive plural (used after many prepositions, adjectives, and numbers)
- Je tam hodně lidí. – There are many people there.
- plné lidí – full of people

In the sentence, lidí is genitive plural because it depends on plné (full of people).

Isn’t turistů a lidí redundant? Don’t tourists also count as people?

Literally yes, tourists are people, but in natural language this is normal and meaningful.

Here turistů a lidí suggests two groups:

turistů – visitors, non‑locals
lidí – other people, e.g. locals, residents, anyone else

It implies: “full of tourists and (other) people”, i.e. crowded with everyone, not just tourists. It’s a stylistic way to emphasize how busy it is.

Could I say Centrum města je dnes večer plno turistů a lidí instead of plné turistů a lidí?

Not exactly. You’d usually say:

V centru města je dnes večer plno turistů a lidí.

Here:

plno is an adverb / indeclinable form meaning roughly “a lot” or “full (of)” in an impersonal structure.
The subject is implicit; you’re basically saying “There are a lot of tourists and people in the city center this evening.”

Your original sentence:

Centrum města je dnes večer plné turistů a lidí.
– grammatically has centrum města as the subject and plné as a normal adjective.

So:

Centrum je plné X. – The center is full of X.
V centru je plno X. – There is a lot of X in the center.

Both are correct; they just use different grammatical patterns.

Why is it je dnes večer plné and not bude dnes večer plné, if we’re talking about this evening?

It depends on the time reference:

If it’s already this evening now, then je dnes večer plné is fine:
“The city center is full this evening” (and it is already evening).
If you’re talking earlier in the day, predicting the future, more natural Czech would be:
- Centrum města bude dnes večer plné turistů a lidí.
  – The city center will be full of tourists and people this evening.

So the present je describes a current state during “this evening”, while bude predicts a future state.

Why is it přijde tam také náš učitel and not náš učitel tam také přijde? Is the word order important?

Both are grammatically correct; the difference is focus/emphasis:

Přijde tam také náš učitel.
Neutral focus is on náš učitel coming as an additional person. Roughly:
“…and our teacher will come there as well.”
Náš učitel tam také přijde.
Starts with náš učitel, so it puts more emphasis on him as the topic. More like:
“Our teacher will also go there.” (compared to someone else who is going)

So the given word order is fine, slightly emphasizing that our teacher is one of the people who will come there too.

What does také do here, and where can it go in the sentence?

Také means “also / too / as well”. In this sentence:

přijde tam také náš učitel
– our teacher will also come there

You can move také to change nuance slightly:

Také náš učitel tam přijde. – also our teacher (maybe others’ teachers as well)
Náš učitel také přijde tam. – slightly odd; usually tam stays close to the verb
Náš učitel tam také přijde. – neutral: our teacher will go there too

The most natural placements are usually before the stressed element (the thing you want to say “also” about): také náš učitel, také tam, etc.

Why is přijde used here and not something like přichází or chodí?

Czech verbs of motion are very aspect‑sensitive:

přijít (přijde) – perfective, one‑time “to come/arrive (will come)”
přicházet (přichází) – imperfective, ongoing or repeated “is coming / keeps coming”
chodit (chodí) – habitual or multi‑directional “to go (on foot) regularly / back and forth”

In the sentence:

přijde tam také náš učitel.
– a single future arrival (he will come there once, at some point this evening).

If you said:

Náš učitel tam chodí. – Our teacher goes there (regularly / habitually).
Náš učitel tam přichází. – could mean “is on his way / is in the process of coming,” or “keeps coming,” depending on context.

So přijde is chosen to express one specific visit in the future.