Áður en við förum heim kveðjum við kennarann, og hann brosir til okkar.

Áður en means before and introduces a subordinate clause.

So:

• Áður en við förum heim = Before we go home

The clause after áður en has normal subject-verb order:

• við förum = we go

Then the main clause follows after that.

This is a very common Icelandic word-order pattern.

In the subordinate clause after áður en, you get normal order:

• við förum = we go

But in the main clause, because something else comes first (Áður en við förum heim), Icelandic uses verb-second word order. That means the finite verb comes before the subject:

• kveðjum við kennarann
• literally: say-goodbye we to-the-teacher

So the structure is:

• fronted element: Áður en við förum heim
• finite verb: kveðjum
• subject: við

This is one of the most important word-order rules in Icelandic.

Förum is the 1st person plural present tense of fara (to go).

So:

• ég fer = I go
• við förum = we go

In this sentence, við förum heim means we go home.

Heim is an adverb meaning home, especially with movement toward home.

So:

• fara heim = go home

This works a lot like English, where we also say go home, not go to home.

A useful contrast is:

• heima = at home
• heim = homeward, to home

So:

• við erum heima = we are at home
• við förum heim = we go home

Because kennarann is the accusative singular definite form of kennari (teacher).

Here, kveðja takes a direct object, and that object is the teacher, so Icelandic uses the accusative form:

• kennari = a teacher / teacher (dictionary form, nominative)
• kennarann = the teacher (accusative singular)

So:

• kveðjum við kennarann = we say goodbye to the teacher

The ending changes because Icelandic marks case on nouns.

It can mean either, depending on context.

In this sentence, the context is:

• Before we go home...

Because of that, kveðjum við kennarann is most naturally understood as:

• we say goodbye to the teacher

In another context, kveðja could mean greet or welcome someone.

Kveðjum is the 1st person plural present tense of kveðja.

So:

• ég kveð = I greet / say goodbye
• við kveðjum = we greet / say goodbye

Because of Icelandic word order, it appears before við here:

• kveðjum við kennarann

But it still means we say goodbye to the teacher.

Because the expression here is brosa til einhvers, meaning smile at someone or smile toward someone.

So:

• hann brosir til okkar = he smiles at us

Also, til takes the genitive case, and okkar is the genitive form of við (we/us).

Compare:

• við = we
• okkur = us (often dative/accusative, depending on use)
• okkar = of us / our / us after til

So til okkar is a prepositional phrase, not a direct object.

Brosir is the 3rd person singular present tense of brosa (to smile).

So:

• ég brosi = I smile
• hann brosir = he smiles

That matches hann (he) in the sentence:

• og hann brosir til okkar = and he smiles at us

Icelandic punctuation does not always match English punctuation.

In English, an introductory clause is often followed by a comma:

• Before we go home, we say goodbye...

In Icelandic, that comma is often omitted in sentences like this:

• Áður en við förum heim kveðjum við kennarann...

The comma before og helps separate the two main parts:

• ...kveðjum við kennarann, og hann brosir til okkar

So the punctuation here is normal Icelandic style.

Yes. That would also be grammatical, but the emphasis would be different.

For example:

• Við kveðjum kennarann áður en við förum heim. = We say goodbye to the teacher before we go home.

In your original sentence, the time clause comes first:

• Áður en við förum heim...

That puts more focus on before we go home. Because that element is moved to the front, the main clause then shows the usual Icelandic inversion:

• kveðjum við

So both are possible, but the original sentence highlights the timing more strongly.