Things you really need:
Canned and bagged food to last a week. And a can opener that doesn't need electricity or all cans need pop tops.
Water to last three weeks. You don't need to buy commercial bottled water. I save milk cartons, wash them, and fill them with tap water. I have 25 bottles lined up in the garage in time order, and recycle the oldest as I add a new one. Or you can empty them and re-fill on your birthday every year.
Cash to last a while. ATMs might not work for several days.
A crank or solar (or combination) cell phone charger.
Never let your gas tank in your car get under 1/2 (thanks Dad for teaching me this!), and if you see a disaster headed your way (only hurricanes are so helpful in this way), fill up your tank as soon as the NHC tells you it's headed your way. Do not assume a Cat 1 hurricane will stay that low, not with rapid intensification the way it's been happening the past decade.
Gas for anything else you have that requires it (chainsaw, generator).
A few big packages of hand wipes in case you need to bathe with them. Hurricanes and earthquakes can result in no water at all coming to you.
Paper plates and napkins. Even if you usually are more "green" than that, you get a pass during an emergency. In a flood situation when water isn't safe to wash dishes in, or an earthquake that leaves you without water, you'll be happy you have them. Paper is better than plastic if you have a safe place to burn it, in case trash pick up doesn't happen for a week. (I have a burn ring at my house.)
Sufficient drugs for a month, prescription or over the counter. Masks for disease and dust reasons. And a decent first aid kit.
Photos you've taken of important documents (birth certificates, insurance, proof you rent or own your home, driver's license, etc) available on your phone and in the cloud.
If you have kids, books and cards and puzzles and other non-electrical entertainments.
Batteries for anything you have that requires them. I use rechargeables, and I always have eight AA and AAA batteries ready to go.
Things you might want:
If you only have an electric stove, a gas grill to cook with. If you don't have power after two days, you may be grilling a lot of food at once and having the neighbors over.
A bug-out bag. Mine, a hat, and sturdy boots are hanging right by my car in the garage. I can grab them and go, and they have plenty. Sturdy boots are crucial because so many disasters leave you with glass, nails, or other hazardous items to walk over. In California, they tell you to keep your boots and a hard hat or bike helmet right under your bed. Good advice! Pre-packed bug-out bags are useful if you need to go to a shelter.
A generator, but only if you understand how to use it safely. I have two relatives with whole-house generators, and I have one in my RV (which also serves as the biggest bug-out bag ever.)
A solar shower. I use Advanced Element's, and I used it twice a week for several years without it ever springing a leak.
Small survival gadgets like a Swiss Army Knife or a multi-tool, a water filter, and a magnesium fire starter (mine has a compass on the end of it). I keep all these in my bug-out bag (which is a backpack), along with toilet paper, a small first aid kit, two bandanas, extra socks, a fleece vest, two "space blankets," plastic bags, a paperback book to read, and a tiny fishing kit in a film cannister).
I also own a rechargeable fan that last four hours on low. There are some that operate off a small solar panel. Think how people in New Orleans would love one of those this week. It's not A/C, spray yourself with water and stand in front of it, and it's better than nothing.
And please, when you hear the words "mandatory evacuation," evacuate!
Most nations and many states and provinces have websites on emergency preparation. In the US, it's ready.gov.