Statt den ganzen Abend im Internet nach Wohnungen zu suchen, gehe ich kurz spazieren, um ruhig zu bleiben.

statt means instead of. Here it introduces an infinitive clause: statt ... zu suchen = instead of searching ...

German often uses this pattern:

statt + [details] + zu + infinitive

So:

Statt den ganzen Abend im Internet nach Wohnungen zu suchen, ...
= Instead of searching online for apartments all evening, ...

You can also hear anstatt instead of statt. They mean the same thing here.

Because this is an infinitive clause with zu.

In German, when you have a structure like statt ... zu, um ... zu, ohne ... zu, the infinitive verb usually goes to the end of that clause.

So the structure is:

• statt ... zu suchen
• um ... zu bleiben

That is why both suchen and bleiben appear at the end of their respective clauses.

Because den ganzen Abend is an accusative expression of time duration.

German often uses the accusative to say how long something lasts:

• den ganzen Abend = all evening
• den ganzen Tag = all day
• eine Woche = for a week

So here den ganzen Abend does not mean it is the direct object of suchen. It tells you the duration of the activity.

Because suchen nach is the fixed pattern used here: nach etwas suchen = to look for something.

Important detail: nach takes the dative.

So:

• die Wohnung → plural die Wohnungen
• dative plural → Wohnungen
  plural nouns in the dative often add -n, but Wohnungen already ends in -n, so it stays the same

So:

nach Wohnungen suchen = to search/look for apartments

A learner should note that English says look for, while German often says nach ... suchen.

im is a contraction of in dem.

So:

• in dem Internet → im Internet

This is very common in German.

Here in is used in the sense of in/on the internet. German normally says im Internet where English often says on the internet.

Because the sentence begins with a fronted clause:

Statt den ganzen Abend im Internet nach Wohnungen zu suchen,

When something comes first in a German main clause, the conjugated verb must still be in position 2. This is the famous V2 rule.

So the main clause is:

gehe ich kurz spazieren

Here is the structure:

1. Statt den ganzen Abend im Internet nach Wohnungen zu suchen
2. gehe
3. ich
4. kurz spazieren

So the verb comes immediately after the introductory clause.

Yes, spazieren gehen is a very common verb combination meaning to go for a walk.

In a conjugated sentence:

• Ich gehe spazieren.
• Er geht spazieren.

The finite verb gehen is conjugated, and spazieren stays at the end.

So in your sentence:

gehe ich kurz spazieren

That is the normal word order for this expression.

Here kurz means something like for a short time or briefly.

So:

Ich gehe kurz spazieren
means: I’m going for a short walk / I’ll go for a brief walk

It does not literally mean the walk itself is physically short in distance; it usually refers to time.

Because um ... zu + infinitive is another fixed German structure.

It expresses purpose:

• um ... zu = in order to ...

So:

um ruhig zu bleiben
= in order to stay calm

This is different from the earlier statt ... zu:

• statt ... zu = instead of doing something
• um ... zu = in order to do something

Both require zu + infinitive.

Because bleiben means to remain/stay, and it fits the idea here: staying calm.

• ruhig sein = to be calm
• ruhig bleiben = to stay/remain calm

So ruhig bleiben emphasizes maintaining a calm state, which is exactly what the sentence wants.

Because ruhig is not being used before a noun here. It is a predicate adjective after bleiben.

In German, adjectives used this way are not inflected.

Compare:

• ein ruhiger Abend → adjective before a noun, so it changes
• Ich bleibe ruhig → predicate adjective, so no ending

So:

ruhig bleiben = to stay calm

Because German normally uses commas to separate these kinds of infinitive constructions.

In this sentence, both of these are infinitive clauses:

• Statt den ganzen Abend im Internet nach Wohnungen zu suchen,
• ..., um ruhig zu bleiben.

So the commas help mark those clause boundaries.

German punctuation is often stricter than English in this area, especially with um ... zu, statt ... zu, and similar constructions.

Yes.

For example:

• Statt eines Spaziergangs arbeite ich weiter.
  = Instead of a walk, I keep working.

But in your sentence, statt introduces an action, so German uses an infinitive clause:

statt ... zu suchen

So both are possible in German:

• statt + noun
• statt + zu-infinitive clause

Your sentence uses the second type.

It is neutral, natural standard German.

Nothing in it is especially slangy or unusually formal. A native speaker could say or write this very naturally.

A few especially common everyday pieces are:

• im Internet
• nach Wohnungen suchen
• kurz spazieren gehen
• ruhig bleiben

So this is a very useful model sentence for real-life German.