Barnen börjar vänja sig vid den nya rutinen i skolan.

Barnen means the children. The base form barn can mean child or children depending on context, but barnen is the definite plural form: the children.

So:

• ett barn = a child
• barn = children / child in some contexts
• barnen = the children

The -en ending is what makes it definite here.

Börjar is the present tense form of börja (to begin).

Because the subject is Barnen (the children), the sentence needs a finite verb, and in Swedish present tense that is often formed with -r:

• att börja = to begin
• börjar = begin / are beginning

So Barnen börjar ... means The children are beginning ...

After börja, Swedish often uses the infinitive with or without att, but in modern everyday Swedish it is very common to leave att out before another verb.

So both can occur:

• börjar vänja sig
• börjar att vänja sig

In this sentence, börjar vänja sig sounds natural and standard.

Vänja sig is a reflexive verb meaning to get used to or to become accustomed to.

The sig is necessary because the verb is used reflexively here. It is similar to saying accustom oneself in English, although English usually just says get used to.

So:

• vänja sig vid något = to get used to something

Without sig, the meaning changes.
For example:

• vänja någon vid något = to accustom someone to something
• vänja sig vid något = to get used to something oneself

Sig is the special reflexive pronoun used for the third person in Swedish when the action refers back to the subject.

Since Barnen are the ones doing the getting used to something, Swedish uses sig:

• Barnen vänjer sig ... = The children are getting used ...

If you used dem, it would mean someone else is doing something to them, not that they are acting in a reflexive way.

Very roughly:

• sig = themselves in a reflexive structure
• dem = them

Because vänja sig normally goes with the preposition vid. This is just the standard pattern of the verb.

• vänja sig vid något = to get used to something

This is something learners usually need to memorize as a unit:

• vänja sig vid
• bli van vid
• vara van vid

It does not translate word-for-word neatly into English, but the whole phrase means get used to.

This is because Swedish uses double definiteness with many definite noun phrases.

When you have:

1. a definite noun,
2. plus an adjective,

you usually use:

• a definite article before the adjective, and
• a definite ending on the noun.

So:

• den nya rutinen = the new routine

Breakdown:

• den = definite article for a common-gender singular noun
• nya = adjective in definite form
• rutinen = routine with the definite ending

Why not den ny rutinen?
Because after den, the adjective usually takes the definite form, which here is nya, not ny.

Because the noun phrase is definite: den nya rutinen = the new routine.

In Swedish, adjectives change form depending on whether the noun is:

• common gender or neuter
• singular or plural
• indefinite or definite

Here, because the phrase is definite, the adjective takes the definite/weak form nya.

Compare:

• en ny rutin = a new routine
• den nya rutinen = the new routine

I skolan means in school or at school, depending on how natural English would express it in context.

Here it most naturally connects to the new routine:

• the new routine in school or
• the new routine at school

Swedish often uses i where English may prefer either in or at, depending on the situation.

So i skolan is very natural Swedish for something happening within the school setting.

Usually i skolan is the natural choice here.

Very generally:

• i skolan often means in/at school as an institution or inside the school context
• på skolan can sometimes be used in certain regional varieties or when focusing more on the school as a workplace/site, but it is not the most standard choice here

So for a learner, i skolan is the form to remember in this sentence.

The basic word order is:

Subject + finite verb + infinitive/reflexive phrase + prepositional phrase

So:

• Barnen = subject
• börjar = finite verb
• vänja sig vid den nya rutinen = verb phrase
• i skolan = prepositional phrase

A useful thing to notice is that Swedish main clauses normally put the finite verb in the second position. In this sentence, the subject comes first, so the verb follows it directly:

• Barnen börjar ...

Yes, very much.

The structure is:

• börjar = are starting / begin
• vänja sig vid = get used to

So:

• Barnen börjar vänja sig vid ... is very close to
• The children are starting to get used to ...

It is a very natural Swedish way to express a gradual change or adjustment.

The present tense verb börjar would become började:

• Barnen började vänja sig vid den nya rutinen i skolan.

That means:

• The children began to get used to the new routine at school.

Notice that the rest of the structure stays the same:

• började vänja sig vid ...

Yes, vänja can feel a little tricky for English speakers.

A few helpful points:

• ä sounds somewhat like the vowel in English bed, but not exactly the same
• nj together gives a sound similar to the ny in canyon or onion
• so vänja is roughly like VEN-ya, but with a Swedish ä

Also:

• vänja sig vid is a phrase worth practicing as a whole, because it is very common in Swedish.