The Start-

To begin Cook's Assistant head into the ground floor of Lumbridge Castle. Find the kitchen and talk to the cook, he will tell you that he has a problem. It's the Duke's birthday today and he needs to bake a cake, this would be simple enough but he forgot to buy the ingredients. Tell him that you will help him out and he will list the needed ingredients: milk, egg, and flour.

Map-

Getting the items-
Now you must gather the ingredients, you can ask the cook where to find them or just refer to this guide. If you don't have a bucket (for the milk) and a pot (for the flour) you can buy them at the general store above Lumbridge Castle. Please refer to the above map for the following instructions.

Milk:

Take your bucket and head over to the dairy cow, select the milk option and you will fill the bucket with milk.

Egg:

Now head to the chicken farm, an egg appears on the ground north of the house. You might have to wait for it to appear, if somebody took it right before you got there.

Flour:

Obtaining flour is slightly more complicated. First head up to the mill, pick some wheat from the field southwest of it before going inside. Once inside climb up to the top floor. Use your wheat on the hopper, then operate the nearby hopper controls. Now climb back down to the first floor and empty the flour bin into your pot.

Back to the castle-
Now that you have your items it's time to give them to the cook. Head back to the castle and talk to him, your character will hand him the ingredients and after a bit of talking the quest will complete.