First, I'd start with the four basic needs for survival:heat, shelter, water, and food (in that order). Heat means a way to stay warm and do cooking. A decent stove and plenty of fuel is essential. Maybe a portable heater as well for the cold months with plenty of fuel. Next, a shelter. This may mean building or finding a bomb shelter. Depending on the disaster, a tent, cave, or cabin in the woods might work. Then food. At least a month's worth, along with a plot of land and seeds and livestock for growing more food. Then water. That's the hard one unless you have land with a good well. For any disaster, I'd say at least a week's worth of water for every person in the household. Have a method of purifying future water when your supply runs low. A good water filter and chloride or chlorine dioxide tablets will work. You may have to go find more water later. If there is a nuclear disaster, don't boil the water to purify it. That will concentrate the radiation.
Next, I'd get spare gas and a full tank of gas for all vehicles. Make sure all vehicles are in working order. Get some spare tires, jumper cables, tire plugs, first aid kit, flashlight, multitool, tool kit, and the most replaced car parts (fuses, bulbs, belts) and keep these items in the trunk. I'd get guns and ammo to protect my family and my stash of supplies. If things get bad, someone will try to steal from you, possibly at gun or knife point.
Then you need a bug out location, a safe place to go if your home is no longer safe or doesn't have the infrastructure you need to survive. You need a bug out bag as well. This is a backpack filled with emergency supplies in case something happens away from your home, your bug out location, and possibly your car (your car breaking down may be the emergency). One should be kept at home, in the car, and at work unless you get accustomed to taking one everywhere with you. A bug out bag should have a way to cook food (stove, pot, matches, magnesium block, Ultimate Survival Blastmatch), disposable handwarmers, a change of clothes, a jacket, an emergency shelter (tube tent, bivy sack, or tent), a day's supply of water (1 gallon), food, knife, multitool, headlight, flashlight, spare batteries, AM/FM/weather radio, a couple or more long range walkie talkies with spare batteries, small first aid kit (including sunblock, bug repellent, Sawyer's Snake Bite Kit, moleskin, chapstick, Sam's Splint, potasium iodide tablets), candles (firestarter and backup for flashlight if there's an EMP blast), and maybe a gun and spare ammo (some places may prohibit this). Make sure the bag is light enough to carry.
Once you've done this, start stockpiling lights, batteries, guns, ammo, knives, multitools, food, water, gas, and camping supplies. In an emergency, you can trade spare supplies for what you have too much of. Learn how to survive by going camping and backpacking. Learn what you can live with and without. Take your family on these trips so they can learn as well. They will be less freaked out during an apocalypse if you taught them how to survive first. Have board games or cards to keep children's minds occupied. Practice hunting and target shooting. Take self defense classes. Get all the knowledge you think you need to survive under any circumstances and get your family involved as well.
When's the party start anyway?
