For short term emergencies like you listed, probably the best thing you can do is have a good first aid kit, and a power supply. I've talked before about camping batteries with a solar recharging mat: basically, a car battery you can recharge. That means you can help people charge their phones, flashlights, batteries, etc.

Water outages are unlikely most places, since those are big, tough structures, whose distribution is underground. However, 5-10 gallons of bottled water, some bleach and/or iodine tablets to make more, and even a lifestraw type emergency backup can't go wrong. OTC meds (Tylenol, Excedrin [also helps avoid caffeine withdrawal] are also good to stock up, and cheap. Canned and other shelf stable foods for a week are also good to store, just keep rotating them so they don't expire.

Long term situations are a whole different bag. So much depends on where you are, what your personal situation is like, that it gets tough. I live in a rural area and have a basement, so my long term is completely different than someone who lives in a city apartment.

For me, I've stocked up on food, water & purification stuff, 2 years worth of basic vitamins, a few months worth of dehydrated food, rechargeable power supply, firewood, ammo (although I still need some more), etc. But I have the space to do that. I've also made friends with my neighbors, which is likely to the key.