Efter forestillingen går vi en tur, og bagefter sætter vi os på en bænk ved springvandet igen.

They are related, but they do different jobs.

• efter is a preposition, so it is followed by a noun phrase:
  • Efter forestillingen = after the performance
• bagefter is an adverb, meaning afterwards / later on:
  • og bagefter sætter vi os ... = and afterwards we sit down ...

So the sentence is not repeating itself unnecessarily. It means something like:

• After the performance, we go for a walk, and afterwards we sit down on a bench by the fountain again.

Danish often uses both if one introduces a time point and the other introduces the next step in the sequence.

Because forestillingen is the definite form: the performance.

In Danish, the definite article is usually added as an ending:

• en forestilling = a performance
• forestillingen = the performance

So Efter forestillingen means After the performance.

This is very common in Danish: instead of a separate word like English the, Danish often attaches the definiteness to the noun.

Forestilling usually means performance, show, or presentation, especially in a theatre or entertainment context.

Depending on context, it could be:

• a theatre performance
• a film showing
• a concert performance
• a staged show

So in this sentence, it most naturally means something like the show or the performance.

This is because Danish follows the V2 rule in main clauses: the finite verb usually comes in second position.

When the sentence starts with something other than the subject, such as a time phrase, the verb comes next, and the subject comes after it.

So:

• Vi går en tur. = We go for a walk.
• Efter forestillingen går vi en tur. = After the performance, we go for a walk.

The same thing happens later in the sentence:

• Bagefter sætter vi os ...

Because bagefter is placed first, the verb sætter comes before vi.

Yes, literally it looks something like that, but idiomatically gå en tur means:

• go for a walk
• take a walk

So:

• Vi går en tur = We go for a walk

Here, en tur often means a short outing, trip, or walk, depending on context. In this sentence, go for a walk is the natural translation.

Os is the reflexive pronoun ourselves, and it is required with sætte sig.

• at sætte sig = to sit down
• literally: to set oneself

So:

• vi sætter os = we sit down

This is a very common Danish pattern. Many verbs use reflexive pronouns where English may not use them in the same way.

Some forms are:

• jeg sætter mig = I sit down
• du sætter dig = you sit down
• vi sætter os = we sit down

Because sætte sig and sidde are not the same.

• sætte sig = sit down
  
  • focuses on the action of moving into a seated position
• sidde = be sitting / sit
  • focuses on the state of already being seated

So:

• Bagefter sætter vi os på en bænk = Afterwards we sit down on a bench
• Bagefter sidder vi på en bænk = Afterwards we are sitting on a bench / we sit on a bench

In this sentence, the idea is that after the walk, they actively sit down, so sætter os fits better.

These prepositions describe different relationships.

• på en bænk = on a bench
  • because you sit on the bench
• ved springvandet = by / near the fountain
  • because the bench is located by the fountain

So the sentence gives both pieces of information:

• where they sit: on a bench
• where the bench is: by the fountain

Springvandet is the definite form: the fountain.

• et springvand = a fountain
• springvandet = the fountain

The ending -et marks definiteness for many neuter nouns.

Here it is definite because it refers to a specific fountain, probably one already known from the situation or context.

Igen usually means again.

So:

• ved springvandet igen = by the fountain again

In this sentence, igen most naturally suggests that they are returning to that place another time, or that this is something they do repeatedly.

Depending on context, English might translate it as:

• again
• once more
• sometimes even back in a looser translation

But the basic meaning is again.

Yes. Danish often uses the present tense for planned or expected future actions, just like English can.

So:

• Efter forestillingen går vi en tur
  can mean After the performance, we’re going for a walk
• bagefter sætter vi os ...
  can mean afterwards we’ll sit down ...

This is completely natural in Danish. You do not always need a special future form.

Yes.

After og, the new clause starts with bagefter, and because that comes first, the finite verb must still be in second position:

• Bagefter sætter vi os ...

Not:

• Bagefter vi sætter os ...

So the structure is:

• bagefter = first element
• sætter = finite verb in second position
• vi = subject

This is a very typical Danish clause pattern.