Learn Hardcore German: Wir kümmern uns um die Kinder.

Yes. The core verb is sich (um etwas/jemanden) kümmern, which is reflexive. That means you must use a reflexive pronoun that matches the subject. With wir, the correct reflexive pronoun is uns. You cannot say just “Wir kümmern um …” in standard German; the reflexive is required. Here, the reflexive pronoun is in the accusative case.

What case does the object after "um" take?

After um, you use the accusative. That’s why it’s um die Kinder. A common mnemonic for accusative prepositions is: durch, für, gegen, ohne, um, bis, entlang.

So is "die Kinder" nominative or accusative here?

Accusative plural. Note that nominative plural is also “die,” so the form looks the same; the preposition um tells you it’s accusative.

Where should the reflexive pronoun go in the main clause?

In a main clause, the conjugated verb is in position 2, and the reflexive pronoun usually comes right after it, before other objects/phrases:

Wir kümmern uns um die Kinder.
With an adverb: Wir kümmern uns heute um die Kinder.

How does this look in a subordinate clause?