Τα παιδιά πιστεύουν στη μαγεία.

Because παιδί is a neuter noun in Greek.

Greek articles must agree with the noun in gender, number, and case:

• το παιδί = the child
• τα παιδιά = the children

So οι would be used for masculine or feminine plural, not neuter plural.

The singular is παιδί.

So:

• το παιδί = the child
• τα παιδιά = the children

This is a very common noun, and learners usually just memorize it as a pair:

• παιδί / παιδιά

Πιστεύουν is the 3rd person plural form of πιστεύω in the present tense.

Since τα παιδιά is plural, the verb is plural too:

• το παιδί πιστεύει = the child believes
• τα παιδιά πιστεύουν = the children believe

You may also sometimes see πιστεύουνε, which is a longer colloquial variant of the same form.

Στη is the contracted form of σε + τη(ν).

So:

• σε = in, to, at
• τη(ν) = the, feminine accusative singular
• σε τη(ν) becomes στη(ν)

In this sentence:

• στη μαγεία = σε τη μαγεία

The final -ν is dropped here, so you get στη, not στην, because the next word begins with μ.

After the preposition σε, Modern Greek normally uses the accusative.

So μαγεία here is in the accusative singular.
With this noun, the nominative and accusative forms look the same:

• nominative: η μαγεία
• accusative: τη μαγεία

That is why the noun itself does not change visibly, but the article does.

Because πιστεύω σε + noun means believe in someone or something.

This is an important pattern:

• πιστεύω σε κάτι = I believe in something
• πιστεύω κάποιον / κάτι = I believe someone / something is true
• πιστεύω ότι... = I believe that...

So:

• Πιστεύω στη μαγεία = I believe in magic
• Πιστεύω την ιστορία σου = I believe your story
• Πιστεύω ότι θα έρθει = I believe that he/she will come

Greek uses the definite article much more often than English does.

With abstract nouns like μαγεία, Greek very often keeps the article, especially in expressions like this one. So στη μαγεία is the normal, natural Greek way to say in magic.

If you leave the article out, it may sound unnatural or stylistically unusual in ordinary speech.

It can mean either, depending on context.

Greek often uses the definite article even when speaking generically:

• Τα παιδιά πιστεύουν στη μαγεία can mean Children believe in magic as a general statement.
• In another context, it could also mean The children believe in magic, referring to specific children.

So Greek uses the article more broadly than English.

No. Greek word order is fairly flexible.

Τα παιδιά πιστεύουν στη μαγεία is the most neutral order:

• subject + verb + prepositional phrase

But Greek can move things around for emphasis:

• Στη μαγεία πιστεύουν τα παιδιά = emphasis on magic
• Τα παιδιά στη μαγεία πιστεύουν = possible, but more marked

Because Greek marks grammatical relationships with articles and endings, word order is freer than in English.

A rough pronunciation is:

ta peh-DHYA pee-STEH-voon sti ma-yee-A

A few helpful notes:

• αι in παιδιά sounds like e in pen
• δ sounds like the th in this
• γ before ε or ι sounds like a soft y-like sound
• The stress falls on:
  • παιδιά
  • πιστεύουν
  • μαγεία

So the rhythm is roughly:

Τα παιΔΙΑ πισΤΕΥουν στη μαγεΙΑ