Jeg tager den ene bog med til skolen, og min søster tager den anden.

Tage ... med is a very common Danish combination meaning take along / bring with you. In a main clause, the particle med is often separated from the verb:

• Jeg tager bogen med. = I’m taking the book along.

So in your sentence, Jeg tager den ene bog med til skolen means something like I’m taking one of the books along to school.

Because Danish normally uses den/det/de before an adjective-like word when the noun is definite.

So:

• en bog = a book
• den ene bog = the one book / the one of the two books

Here ene works in a way similar to an adjective, so Danish uses:

• den ene bog
• compare: den røde bog = the red book

Also notice that the noun stays in its basic form after den + adjective-like word:

• den ene bog
• not den ene bogen

In this sentence, den ene means one of the two. It is usually paired with den anden = the other one.

So:

• den ene bog = one of the two books
• den anden = the other one

This pair is very common in Danish when talking about two people or things.

Not necessarily. Den ene does not automatically mean the first one in order. It just means one of the two.

If you really want to say the first, Danish normally uses:

• den første

So:

• den ene og den anden = one and the other
• den første og den anden = the first and the second

Because Danish, like English, often leaves out a noun when it is already understood from the context.

So:

• min søster tager den anden
  means
• my sister takes the other one

The word bog is omitted because it is already clear that we are still talking about books.

You could say min søster tager den anden bog, but it is less natural here because the noun does not need to be repeated.

Because til shows movement toward a place:

• til skolen = to the school

By contrast, i usually means in or inside:

• i skolen = in the school

So if someone is going there, til is the natural choice. If someone is already there, i may be used depending on the context.

This is a very common difference between English and Danish.

English often says:

• to school
• at school

Danish often uses the definite form:

• til skolen
• på skolen or i skolen, depending on meaning

So even though English has no article here, Danish often does. This is just one of those places where Danish and English structure things differently.

Also note that Danish has the idiomatic expression:

• i skole = in school / at school

But that is a slightly different expression from til skolen, which focuses more on going to the school.

Because the sentence contains two full clauses, each with its own subject and verb:

• Jeg tager den ene bog med til skolen
• min søster tager den anden

In Danish, it is standard to put a comma between coordinated main clauses like these, even when they are joined by og.

Yes, that is possible.

• min søster tager den anden = my sister takes the other one
• min søster tager den anden med = my sister takes the other one along

The version without med is perfectly natural because the idea of taking the other book is already clear from the context. Adding med makes the take along idea more explicit.

Because bog is a common-gender noun in Danish, and common-gender nouns use den.

• en bog = a book
• therefore: den ene bog, den anden

If the noun were neuter, you would use det instead:

• et hus = a house
• det ene hus, det andet

So the form depends on the gender of the noun being referred to.

Because possessives like min, din, hans, hendes usually replace the article.

So Danish says:

• min søster = my sister

not:

• den min søster

This works much like English: you say my sister, not the my sister.