Learn Hardcore German: An Silvester schauen wir zusammen das Feuerwerk über der Stadt. - On New Year’s Eve we watch the fireworks over the city together.

With days you normally have an article, so you say:
- am Montag (= an dem Montag)
- am 1. Januar
With names of holidays, German often drops the article, so there is no dem to contract:
- an Silvester
- an Weihnachten
- an Ostern

Because Silvester here is used without an article (an Silvester, not an dem Silvester), you can’t say am Silvester in standard German.
Colloquially some people do say am Silvester, but an Silvester / zu Silvester is what you will usually see in correct written German.

Could I also say Zu Silvester instead of An Silvester?

Yes.

Both are correct, and both are common in contemporary German:

An Silvester schauen wir…
Slightly more formal/literary; literally “on New Year’s Eve…”.
Zu Silvester schauen wir…
Very common in everyday speech; literally “for / at New Year’s Eve…”.

In meaning there is no real difference in this context. You can safely use zu Silvester in casual conversation.

What exactly does Silvester refer to? Is it New Year’s Day?

No. In German:

Silvester = New Year’s Eve (the evening of December 31st)
Neujahr = New Year’s Day (January 1st)

So the sentence means “On New Year’s Eve we watch the fireworks together over the city”, not on January 1st.

Why is Silvester capitalized?

Silvester is capitalized because:

It is a noun (all nouns are capitalized in German).
It is also the name of a holiday, which makes it a proper noun (like Weihnachten, Ostern).

Even if you didn’t know it was a holiday, its capital letter tells you it is a noun.

What’s the difference between schauen and sehen here?

Both relate to “seeing”, but:

sehen = to see (more general, more neutral)
schauen (often anschauen) = to look at, to watch (more active, intentional)

In this context, schauen (das Feuerwerk) is like English “watch the fireworks”.
You could also say:

Wir sehen (uns) das Feuerwerk an.
Wir gucken das Feuerwerk. (colloquial, regional: gucken or kucken)

All are understandable, but (an)schauen is a natural choice for “watching” something.

Why is the verb schauen in second position: An Silvester schauen wir… and not “An Silvester wir schauen…”?

German main clauses follow the V2 rule: the conjugated verb must be in second position, no matter what comes first.

If the subject comes first:
- Wir schauen an Silvester das Feuerwerk.
If a time expression comes first:
- An Silvester schauen wir das Feuerwerk.

In An Silvester schauen wir…, the positions are:

An Silvester = first element (time phrase)
schauen = conjugated verb (must be second)
wir = subject
zusammen das Feuerwerk über der Stadt = rest of the sentence

“An Silvester wir schauen …” breaks the V2 rule, so it’s wrong.

Why is zusammen placed after wir? Could I say Wir schauen zusammen das Feuerwerk?

Yes, you can say both:

An Silvester schauen wir zusammen das Feuerwerk…
An Silvester schauen wir das Feuerwerk zusammen…

Both are correct and common. Differences:

…wir zusammen das Feuerwerk…:
Puts a bit more emphasis on “we together” as a group activity.
…wir das Feuerwerk zusammen…:
Feels slightly more neutral; the adverb just sits near the verb phrase.

In many everyday sentences, zusammen appears right after the subject or after the verb:

Wir zusammen schauen… (unusual)
Wir schauen zusammen… (very natural)
Wir schauen das Feuerwerk zusammen. (also natural)

So yes: Wir schauen zusammen das Feuerwerk is perfectly fine.

Why is it das Feuerwerk and not just Feuerwerk without an article?

In German, countable singular nouns usually need an article unless they’re used in a very generic or abstract way.

Here:

Feuerwerk is a neuter noun: das Feuerwerk
It is a specific fireworks display (the one in the city on New Year’s Eve), not fireworks in general.

So you say:

Wir schauen das Feuerwerk. = We watch the (specific) fireworks (display).

If you say Wir schauen Feuerwerk, it sounds unnatural.
You could, however, talk about fireworks in general:

Feuerwerk ist an Silvester sehr beliebt.
(“Fireworks are very popular on New Year’s Eve.”)

Why is it über der Stadt with der, not über die Stadt with die?

The preposition über can take either dative or accusative:

Dative = location (where something is)
Accusative = direction / movement (where something is going)

Here we’re talking about where the fireworks are located in the sky: above the city, not moving to above the city.

über der Stadt → dative = where? above the city (location)
über die Stadt → accusative = over the city (movement across/over it)

Because Stadt is feminine:

Nominative: die Stadt
Dative: der Stadt

So über der Stadt is “over the city” in the sense of a fixed location in the sky.

Could I say über die Stadt in any context?

Yes, but the meaning changes. Über die Stadt with accusative suggests movement:

Die Flugzeuge fliegen über die Stadt.
= The planes fly over the city (across it, from one side to the other).

In the fireworks sentence, you’re not focusing on the fireworks moving over the whole city, but on them being up in the sky above the city, so dative (über der Stadt) is more natural.

What case is das Feuerwerk in, and why?

Das Feuerwerk is in the accusative case because it is the direct object of the verb schauen:

Wer? → wir (nominative subject)
Was schauen wir? → das Feuerwerk (accusative object)

Since Feuerwerk is neuter:

Nominative: das Feuerwerk
Accusative: das Feuerwerk

The form looks the same; only in masculine nouns do you see a clear change (der → den).

Could I say Wir sehen das Feuerwerk über der Stadt instead?

Yes. That’s grammatically correct and idiomatic.

Wir sehen das Feuerwerk über der Stadt.
Wir schauen (uns) das Feuerwerk über der Stadt an.
Wir schauen das Feuerwerk über der Stadt. (as in the original)

sehen is more neutral (“we see”), (an)schauen is more like “we watch”. In everyday use both are fine here; many speakers might even prefer sehen or ansehen.

Why is it An Silvester and not Im Silvester?

Because of how German uses time prepositions:

an is used for points in time, especially days and dates:
- an Silvester, an Weihnachten, am Montag, am 3. Mai
in / im is used for longer periods:
- im Winter, im Januar, in diesem Jahr

Silvester is treated like a day (a specific point in time), so the natural preposition is an, not in.

Is there anything special about starting the sentence with An Silvester?

Starting with An Silvester simply emphasizes the time. It’s a very common pattern in German:

An Silvester schauen wir… (time first)
Wir schauen an Silvester… (subject first)

Both have the same meaning. Putting the time phrase at the front is often used to set the scene, especially in storytelling or descriptions of habits.