Invest in a top notch NOAA radio -there are many available that you can charge by cranking, solar power, USB, or the car AND that can charge your phone or laptop for you if it's got enough charge in it- and make sure that it's properly set to the right stations for your area with alarms turned on. Put it somewhere central so you'll hear it.
If you have camping gear, which doubles as good emergency gear, make sure to take it out regularly, seal the seams of your tents, and waterproof it in case you need it. Tents are even useful indoors if you lose power in a cold climate (a tent is a lot less space to heat up than a room).
Invest in appropriate supplies for the temperature extremes in your area. For us that's layers of thermal long underwear and we also got mummy sleeping bags that are highly rated for each of the kids plus a set that zip together for ourselves.
If you shop at a warehouse store like Costco, keep an eye out for large, sealable plastic buckets that are labeled #1 plastic on the bottom. These are food and water safe. We save the buckets from our cat food and when we expect bad weather, we fill them with water. They don't seal very tight once used, so they aren't very useful for keeping things dry.
Stockpile LED lights. You can never have too many LED lightsticks and lanterns. I also stockpiled replacement batteries. It makes an enormous difference having safe light with no risk of burning down the building. With lots of lights we kept the family sane and entertained by playing the piano (which brought over neighbors with instruments of their own), reading out loud, doing puzzles, playing card and board games.
Make sure you have good boots, excellent wool socks, moleskin, and backpacks. Ideally practice their use. We live miles from the nearest supermarket, we ended up with a lot of guests once our power came on, and we needed more food, but the roads everywhere were closed, so we strapped packs on the kids, the guests, and ourselves, and we hiked into town, carrying back 50+ pounds each (well, on the adults). It turned out to be a great memory, but my boots were old and fell to pieces, so I was in flip flops.
If you have small children and live anywhere near water, I'd have life vests and child carriers such as slings or wraps (not strollers) in case you need to evacuate by water. Turns out you can carry a large toddler on your back with an improvised silk wrap. Don't know about less sturdy fabrics.