Learn Russian: Дети хотят видеть своих кумиров.

Because кумир (idol) is an animate noun (it refers to a person). In the plural, the Accusative case for animate nouns is identical to the Genitive plural. The verb видеть (to see) requires the Accusative case for its direct object, so both the noun (кумиров) and its modifying pronoun (своих) take this "Genitive-looking" form.

Why use "своих" for "their" instead of "их"?

In Russian, when a possessive word refers back to the subject of the sentence (here, Дети, the children), you must use the reflexive possessive свой. If you used их here (их кумиров), it would mean the children want to see someone else's idols, not their own.

Is "дети" a regular plural noun?

No, it is highly irregular! The singular word for "child" is ребёнок. When it becomes plural ("children"), the word changes completely to дети. Here it is in the Nominative case because the children are the subject doing the wanting.

I remember the verb "to want" (хотеть) is irregular. Is "хотят" expected here?

Yes, but it is tricky! Хотеть is a "mixed" verb. In the singular (I, you, he/she), it uses standard "Type 1" endings (like он хочет). But in the plural (we, you, they), it switches to "Type 2" endings. That is why the "they want" form is хотят (using the -ят ending) instead of "хотют".