Every day for breakfast I feed my children WITH/ON milk.

With food, NO preposition needed. "Everyday for breakfast I feed my children milk."

I used to feed myself WITH/ON pasta.

No, you don't really feed yourself. Better: "I used to eat pasta."

Every day I feed milk to my children.

Correct.

Some common usages of "feed." 

I feed my cat every night at 7 p.m. I feed my cat good food. (no preposition)

Feed the baby! He's crying and hungry!

This is a zoo. Don't feed the animals any food.

We need to find ways to feed the hungry and clothe the poor.

You should feed your soul with good books and good music. (Here, the preposition "with" is appropriate - non-food items.)

The prisoners were fed bread and water every day for ten years.

The politicians feed us lies. (Same structure as "The parents feed their child food.")

Hi Simone,

If you are eating something, you "feed on [something]".

If you give food to someone, you "feed [something] to [someone]" or you ""feed [someone] [something]". The key concept here is "give", and "feed" follows the same grammatical structure.

When we use "with", it means "using".