Wij schrijven het kind in bij het kinderdagverblijf.

By itself, schrijven means to write.

But in this sentence, the real verb is inschrijven, which is a separable verb:

• inschrijven = to register / enroll
• schrijven = to write

So:

• Wij schrijven het kind in. = We register/enroll the child.
• Wij schrijven een brief. = We write a letter.

The little part in changes the meaning completely.

Because inschrijven is a separable verb. In a normal main clause, Dutch splits it:

• Wij schrijven het kind in.

Here:

• schrijven is the finite verb, so it goes in the usual verb position
• in moves later in the sentence

This is very common in Dutch.

Compare:

• Ik schrijf me in. = I register myself.
• Wij schrijven het kind in. = We enroll the child.

But when the verb is an infinitive or comes at the end of a subordinate clause, it stays together:

• het kind inschrijven = to enroll the child
• omdat wij het kind bij het kinderdagverblijf inschrijven = because we enroll the child at the daycare center

That is the normal word order for a separable verb in a main clause.

A very neutral pattern is:

• subject + finite verb + object + separable part + other information

So:

• Wij
  • schrijven
    • het kind
      • in
        • bij het kinderdagverblijf

English keeps the verb together, but Dutch often splits it like this.

Here bij means something like with / at an institution or organization.

With inschrijven, Dutch often uses bij to show the place or institution where someone is being registered:

• ingeschreven bij de gemeente = registered with the municipality
• ingeschreven bij een school = enrolled at a school
• ingeschreven bij het kinderdagverblijf = enrolled at the daycare center

So bij does not mainly mean physical location here. It is more about being on the records of that institution.

Both wij and we mean we.

The difference is mainly style and emphasis:

• we = more common in everyday speech
• wij = a bit more emphatic, careful, or formal

So this sentence could also be:

• We schrijven het kind in bij het kinderdagverblijf.

That would sound very natural too.

Because both nouns are het-words.

• kind is a het-word → het kind
• kinderdagverblijf is also a het-word → het kinderdagverblijf

For compound nouns, Dutch usually takes the gender/article from the last part of the compound. The last part here is verblijf, and verblijf is a het-word, so:

• het verblijf
• het kinderdagverblijf

Because Dutch normally writes compound nouns as one word.

So:

• kinder
  • dag
    • verblijf → kinderdagverblijf

This is completely normal in Dutch. English often separates or hyphenates similar expressions, but Dutch prefers one combined word.

You can think of it roughly as:

• kinder = child/children
• dag = day
• verblijf = stay/residence/place of staying

Together it means daycare center.

That is just how many Dutch compounds are formed. The first part of a compound does not always look exactly like the regular plural.

So although the normal plural is:

• kind → kinderen

the compound uses:

• kinderdagverblijf

This kinder- form is common in compounds, for example:

• kinderboek = children's book / child’s book
• kinderarts = pediatrician
• kinderkamer = children’s room / nursery

So this is something you mostly learn as part of the word.

Both are possible, but they mean slightly different things in tone.

• het kind = the child
  This is neutral and works well in example sentences or when the child has already been identified.
• ons kind = our child
  This is often more natural if the speakers are the parents and are talking about their own child.

So in real life, parents might very well say:

• We schrijven ons kind in bij het kinderdagverblijf.

But het kind is still perfectly grammatical.

Often yes, but the nuance can be a little different.

• aanmelden = to sign up / report / apply
• inschrijven = to register / enroll officially

For a daycare center, both may be used depending on context:

• We melden het kind aan bij het kinderdagverblijf.
  This can sound like we sign the child up.
• We schrijven het kind in bij het kinderdagverblijf.
  This sounds more like we officially enroll/register the child.

In everyday use, the difference is not always huge, but inschrijven often feels a bit more formal or administrative.

Yes, but then Dutch usually uses a reflexive pronoun:

• Ik schrijf me in. = I register myself.
• Zij schrijft zich in. = She registers herself.

In your sentence, the verb is used with a direct object instead:

• Wij schrijven het kind in.

So:

• zich inschrijven = to register oneself
• iemand inschrijven = to register someone

That is a very useful pattern to remember.