First of all, the correct translation for

What do you want to eat?

is

Qu’est-ce que vous voulez manger?

or

Qu’est-ce que tu veux manger?

if you know the person and can use the “tu” pronoun. If you want to be more polite, then you would say:

Que voudriez vous manger?

but then that’s more akin to:

What would you like to eat?

With respect to the answer, I would translate

Cheerios with milk

as

Des cheerios avec du lait

and not:

Cheerios avec du lait

I believe the “des” is necessary for the answer to be in correct French. Likewise, if you say

I would like cheerios with milk

that would also require the “des”:

Je veux/voudrais des cheerios avec du lait

I can’t think of any circumstance where you wouldn’t use du/des/de la.

When you are talking about cheerios, you find yourself in exactly le same context as when talking about peas. In French each of the little rings is a cheerio;

L’encyclopédie libre … première fois le bol de céréales garni de fraises, accompagné d’un Cheerio sur le « i » de la marque.

As when you say “I eat peas.” in English, in French you have to say “Je mange des petits pois.”, then, when you say “I eat Cheerios.” in English (you don’t eat the brand then …), in French you have to say “Je mange des cheerios.”. It should be as simple as that (unless there are people that need unecessary complications).