Dried potatoes come in flakes, sliced and pearls. Your family may not like flakes, but if they don’t have fresh then they probably will eat flakes if you add things to it. If you use can chicken stock instead of water or add powdered butter to them and put spices on them and they will taste almost just like real potatoes. A variation is powdered ranch dressing or cheese. They can be put up the same way that rice is. One lbs of dry potatoes is equal to about 10 lbs of fresh and you do not have to worry as much about spoilage.
On eggs, you can break them into a bowl, lightly salt, stir and place in ice cube trays to freeze. After they are frozen you can place into plastic bags, double bagged is best and put back into freezer. These will last for a few months. If you are concerned about the power going off, you need to look at either having chickens or getting powder eggs. Again, they won’t taste like the fresh but if you make them into omelets then they will taste fine. On any of this if your children do not like say omelets, start getting them use to it by adding bacon and cheese and work up to more veggies. Also, if you can purchase some powder eggs soon and start getting them use to it, powdered milk would be the same. One thing about powdered and dried food, you don’t have to worry about spoilage of fresh and having to throw something out. Also you are not taking up space by storing egg shells or orange peels, etc.
On flour, cake mixes, cornmeal, pancake mix, and things like this, there are fats in the mix and flour and they will not store long without refrigeration. Instead you will need to find whole wheat, barley and rye, whatever you use to make bread. This can be stored like the rice, you can even take and put the whole wheat in a non-garbage bag liner and put in a large, new plastic garbage can on rollers preferably so you can move it around to where you will be cooking. Baking powder, baking soda, and cornstarch will be needed if you bake. Yeast does not last too long, only about a year if you have it in a refrigerator. So, look into making sourdough for your bread, it lasts longer than yeast.
With sugar, as long as it is kept clean and dry, it is supposed to have an indefinite shelf life. If you want brown sugar just add molasses to white sugar, because brown sugar will not last as long. You can also put up honey. Molasses does have a long shelf life, however, again if it changes color, has a different odor, has mold or taste different throw it out. Molasses would be good to have for cookies (comfort food for children).
Oil will not last long either, without refrigeration and in air tight containers. There is shortening and butter or margarine powder that can be purchased for longer storage time. Also there is baking cocoa, dough enhancer, buttermilk powder, cottage cheese and sour cream. There is also freeze dried cheeses, yogurt bites and even ice cream bars.