Vi flyttar in i den nya lägenheten på lördag.

Because Swedish often uses the present tense for planned future events when the time is clear from the context.

Here, på lördag tells you when it will happen, so Vi flyttar in ... på lördag naturally means a future plan.

This is similar to English sentences like We move in on Saturday, although English often prefers We’re moving in on Saturday.

You could also say Vi ska flytta in på lördag, but the simple present is very common and natural here.

Flytta means move or relocate.
With the particle in, it becomes flytta in, which means move in.

So:

• flytta = move
• flytta in = move in
• flytta ut = move out

This is a particle verb, which is very common in Swedish.

Because they do different jobs:

• in belongs to the verb flyttar in = move in
• i den nya lägenheten tells you where they are moving

So:

• flyttar in = move in
• i den nya lägenheten = into/in the new apartment

Even though it may feel repetitive to an English speaker, it is normal Swedish.
You can think of it a bit like English move in to the new apartment.

Also, i after a motion verb can often correspond to English into, not just in.

Yes, absolutely — if it is already clear where you are moving in.

Then i den nya lägenheten is simply omitted because the destination is understood.

So:

• Vi flyttar in på lördag = We’re moving in on Saturday
• Vi flyttar in i den nya lägenheten på lördag = We’re moving into the new apartment on Saturday

The longer version just gives more information.

Because Swedish usually uses double definiteness when a noun is definite and has an adjective before it.

That means you normally get:

• a definite article before the adjective: den
• the adjective in its definite form: nya
• the noun with its definite ending: lägenheten

So:

• lägenheten = the apartment
• den nya lägenheten = the new apartment

This pattern is very important in Swedish:

• den stora bilen = the big car
• det nya huset = the new house
• de gamla böckerna = the old books

Because lägenhet is an en-word.

In Swedish, nouns have grammatical gender:

• en words take den
• ett words take det

So:

• en lägenhet = an apartment
• den nya lägenheten = the new apartment

Compare:

• ett hus = a house
• det nya huset = the new house

Because adjectives change form in Swedish.

In a definite phrase like den nya lägenheten, the adjective usually takes -a.

Compare:

• en ny lägenhet = a new apartment
• den nya lägenheten = the new apartment

So the adjective changes because the noun phrase is definite.

This -a form is also used in the plural:

• nya lägenheter = new apartments
• de nya lägenheterna = the new apartments

It means on Saturday, usually understood as this coming Saturday or Saturday, depending on context.

Swedish uses på with days of the week in this kind of expression:

• på måndag = on Monday
• på fredag = on Friday
• på lördag = on Saturday

A few useful comparisons:

• på lördag = on Saturday / this Saturday
• i lördags = last Saturday
• nästa lördag = next Saturday
• på lördagar = on Saturdays / every Saturday

Yes. You can say:

På lördag flyttar vi in i den nya lägenheten.

That is completely natural.

But when something other than the subject comes first in a Swedish main clause, the finite verb must come second. This is the V2 rule.

So:

• Vi flyttar in ... på lördag.
• På lördag flyttar vi in ...

Notice that in the second version, flyttar comes before vi.

Yes, you could, but the nuance is slightly different.

• flytta till focuses on moving to a place
• flytta in i focuses more on moving into it

So flyttar in i den nya lägenheten is especially natural when talking about taking up residence in the apartment itself.

In many everyday situations, both versions would be understood, but flytta in i is often the most precise choice here.