If you live city centre, you need to think differently to someone in the depths of the countryside.
If your current systems fail, what is your backup?
If you heating is gas, centralised boiler, electric radiators ...whatever it is - how would you manage if it went off? Do you have any kind of secondary heating (even if expensive to run so you don't normally use it, a gas fire? An oil filled radiator you could plug in to heat 1 room? Or at least a couple of spare blankets and a hot water bottle each?)
And how can you conserve heat in the space you are using - good curtains/blinds on windows, block any spots around windows where you get draughts, draught excluders around doors, film on windows to bounce heat back in or foil lining behind radiators to bounce heat back into rooms than losing it into walls, etc are all generally relatively low cost, going upwards to double or triple glazing etc if fund allow.
And if you have to deal with a major outtage, keep general activity to 1 or 2 main rooms that you heat and close off the other rooms. Wear extra layers. Have cosy socks on your feet. Snuggle under blankets when sitting for a long period.
Water - you should have a few litres of drinking water per person, in case the supply is unexpectedly cut. Just 1 5l bottle, or a couple of 2l bottles would be useful and cheap. And think about some way to collect or store water if you get some notice (we used to fill the bath and use a bucket from that for filling toilets growing up as we had a number of relatively planned outtages, but in a flat that might be more like having 1 or 2 large pots or 1 bucket to fill). If you have a laundry basket, anyway, having one which is solid rather than has lots of holes, can be really useful for various jobs like holding water in a supply outtage, catching drips in a leaking ceiling situation, bathing a dog, carrying liquids or small items, even the floorwashing bucket that gets washed out after use - not just large items that won't fall through the holes.
Electricity goes out? Do you have good natural light when it is daytime? Do you have any kind of non-electric lighting for hours of darkness - torches, lamps, lanterns, candles etc. They might be naked flame (oil lamps/lanterns, candles, nightlights) or batteries or windup. Some can be rechargeable, whether from the socket or a power bank you have previously charged up. Rechargeable batteries can also be useful. Any naked flame solutions need to be done safely - proper glass globes/surround for lanterns/lamps, candles in proper holders, never leaving them burning unattended, having a way to put out fires, store any fuel for lanterns safely etc.
Cooking? Do you have any other way to cook if the power went out or gas supply was interrupted? A 1 ring gas cannister camping stove can be very cheap, especially as shops sell off summer camping gear around now to make space for winter stock - just have the small suitcase package and a few cannisters (like a large can of shaving foam) somewhere, as long as you can use it outdoors (on a patio or balcony is fine - just not indoors for carbon monoxide reasons). Or do you have a BBQ you use in summer? Just make sure you always have fuel for it (charcoal or gas depending on your type) as that can be used to cook even in the depths of winter. If you have any outside space you can use safely, even the disposable BBQs might be handy - as well as the normal grilling of meat/veg etc, you can put a small pot or pan on it to cook liquids, make up tin foil parcels for other types of cooking, put smaller pieces of food onto skewers to manage etc.
And also having some things that don't need cooking, or just some hot water, might be useful also. (Even if your cooking is off, you might be able to get a neighbour or local coffee shop etc to fill a flask with hot water for you, especially if you have the flask yourself - and you can then have soup, hot drinks, couscous, pot noodle type meals available). (Pot noodle type meals have really improved in recent years - there is a wide range of pasta/rice/noodle 1 person meals, pots of porridge, etc available in normal supermarkets now, as well as the more expensive long-life camping dehydrated meals of all sorts in outdoor stores).
And also think about meals that don't require cooking - cold beans, sandwiches, tinned tuna, eggs you have boiled before power went out or added to a pot while cooking potatoes/rice for dinner on your camping stove/bbq etc, pre-cooked bacon/sausages, salads, tinned corn, etc.
And think about what's in your fridge and freezer if power goes out? What needs to be used up quickly and what will last longer even without staying very cold? Don't open the doors of refrigeration appliances often or for long, to keep as cold as possible. If there is ice in shops, that could be useful to buy a bag and put into a fridge/freezer to keep it colder. Or to use in a coolbox if you have a prolonged outtage and you want to just keep a smaller space for a few key items cold, having emptied out the fridge.
Bug out bag to grab in case you need to leave in a hurry - others have already covered this.
And having some cash on hand.
Also - it's not all about mere survival. Morale is important in any situation where you are dealing with emergencies and difficulties, even if you are prepared for them. So the ability to make a cup of tea if possible is huge. Having some favourite beverage may be very useful (not a huge stash of crates of wine - but the ability, once you have dealt with the emergency and can relax, to have a glass or a beer etc, may be good). A spare bar of special chocolate to nibble a square. Comforting drinks like tea, hot chocolate, coffee etc are great. DCs may well enjoy a hot squash (make up squash with warm rather than cold water) if it is cold - both to warm them up and feel better at once.
And entertainment that YOU enjoy which doesn't need power (or at least, not directly connected to a socket). A power bank to recharge an ipad to watch downloaded movies. A board game or deck of cards. Good book or book of puzzles. Crafts you enjoy and can work on. Having a radio that works off batteries or can be wound up can be good for general morale (just listening to music) but also keeping informed.
Slightly more out there ideas which come from camping etc could be having a small solar panel that would recharge your power banks by day; a camping shower (basically a strong black plastic bag which heats up water inside it when placed in sunshine and has a hose/nozzle to shower with, would be useful for washing yourself or washing dishes if you have a sunny windowledge that you can put it) etc.
I keep seeing ideas for heating small rooms using tealights and a terracotta plant pot - but if you have tealights and a warming tray (like the Chinese restaurants used to put shared meals in the middle of the table), you could use those to heat a tin of beans or small pot of water or soup etc - it will not be quick, it may not get piping hot, but if you are in a very bad situation, it would definitely be better than nothing and that kind of dish might be something you would have anyway for nice dinner parties. Or you could put something together using a baking tin to put the tealight into on a heatproof mat, and a baking/cooling rack over, to MacGuyver your own version - KEEP IT IN A SAFE PLACE WHERE IT CANNOT BE KNOCKED AND WELL AWAY FROM DCs!!! And make sure you have a fire extinguisher/blanket to hand.