Jeg tager den nye bog med til biblioteket, men min søster vil hellere læse den gamle.

Tage med is a very common Danish verb expression meaning to take along or to bring.

In a main clause, Danish usually puts the finite verb early in the sentence, so the particle med gets separated:

Jeg tager den nye bog med.
= I’m taking/bringing the new book along.

So tager and med belong together in meaning, even though they are split apart by word order.

You can also see them together in an infinitive:

Jeg vil tage den nye bog med.

In Danish, when a noun is definite and has an adjective before it, the usual pattern is:

den/det/de + adjective + noun

So:

den nye bog
= the new book

This is different from the simple definite form without an adjective:

bogen
= the book

So:

• bogen = the book
• den nye bog = the new book

You do not normally say den nye bogen in standard Danish.

In Danish, adjectives often take -e in the definite form.

That is why you get:

• den nye bog
• den gamle

Compare with the indefinite forms:

• en ny bog = a new book
• en gammel bog = an old book

So the -e here is a normal sign that the adjective is in a definite phrase.

Because Danish, like English, can leave the noun out when it is already understood from context.

Here, den gamle means:

• the old one
• or more specifically, the old book

Since bog was already mentioned, repeating it is unnecessary.

So:

min søster vil hellere læse den gamle
means
my sister would rather read the old one

Hellere means rather.

So:

vil hellere læse
means
would rather read or prefers to read

Literally, vil is the present tense of ville, but in phrases like this, English often translates it more naturally as would rather:

Min søster vil hellere læse den gamle.
= My sister would rather read the old one.

A close alternative would be:

Min søster foretrækker at læse den gamle.
= My sister prefers to read the old one.

But vil hellere is very common and natural.

Biblioteket is the definite form of bibliotek.

• et bibliotek = a library
• biblioteket = the library

So:

til biblioteket
means
to the library

The noun bibliotek is a neuter noun, which is why its definite ending is -et.

Til is used for movement toward a destination:

Jeg tager den nye bog med til biblioteket.
= I’m taking the new book to the library.

If you want to say that someone is already there, Danish often uses på:

Jeg er på biblioteket.
= I’m at the library.

I biblioteket is more literal, meaning inside the library building, and is less idiomatic when you simply mean at the library.

So a useful contrast is:

• til biblioteket = to the library
• på biblioteket = at the library

This follows normal Danish main-clause word order.

The finite verb usually comes in the second position:

• min søster = subject
• vil = finite verb
• hellere = adverb
• læse = infinitive

So:

Min søster vil hellere læse den gamle.

This is the natural order.

It is similar to other modal constructions:

• Jeg vil gerne læse.
• Hun kan godt forstå det.

The modal verb comes early, and the infinitive stays later in the clause.

Because men is a coordinating conjunction, like but in English. It links two main clauses.

So after men, the next clause still has normal main-clause word order:

..., men min søster vil hellere læse den gamle.

That is different from structures where some other element comes first and causes inversion, for example:

I dag tager jeg bogen med.
Here I dag comes first, so tager comes before jeg.

But after men, you just start a new main clause normally.

Because possessives such as min, din, hans, vores already make the noun definite in meaning.

So Danish says:

• min søster = my sister
• min bog = my book

You do not add den in front of a possessive like that.

So min søster is the correct and natural form.

Yes. Tager is present tense, but Danish present tense often covers several meanings that English separates more clearly.

So:

Jeg tager den nye bog med til biblioteket

can mean, depending on context:

• I take the new book to the library
• I’m taking the new book to the library
• sometimes even a planned near-future idea

English often uses the progressive I’m taking, but Danish usually just uses the simple present form.

It depends on the grammatical gender of the noun.

In this sentence:

• bog is a common-gender noun: en bog
• bibliotek is a neuter noun: et bibliotek

That gives:

• den nye bog = the new book
• det nye bibliotek = the new library

In the actual sentence, biblioteket is the definite form of et bibliotek, so you do not see det, but the gender is still neuter.

A useful summary:

• common gender: en bog, den nye bog, bogen
• neuter: et bibliotek, det nye bibliotek, biblioteket