Whether the power is off for a short time, or quite a bit longer, it is important to think now about how to handle your basic needs.
All emergency kits should have a good quality flashlight, and spare batteries. A battery powered radio will be good to keep tabs on weather or news, have spare batteries for the radio as well. A small solar panel can be used to charge rechargeable batteries for long outages. If it is possible for you to get a small generator and store gas, do it, but remember that you cannot store enough gas for long outages.
Solar yard lights can also be used for emergency lighting. I have found a very basic solar light at the big box store for $2.00. Alone it will not provide much more light other than keeping you from stubbing your toe on furniture, so having a few is good. Even a few of them will not allow you enough light for specific tasks, but it will definitely beat being in the dark. Place them outside during the day, and bring them inside at night. I have modified a few that I have, by adding a micro switch to not allow them to run continuously and preserve the charge.
Candles should also be kept available in your supplies, along with matches or a lighter. Tea light candles are another good source for light. They are small, inexpensive, come many to a pack and can last for 2-3 hours.
By using a small 150W inverter that plugs into your cars cigarette lighter, and an extension cord, you can power small devices and lighting inside your home. Purchase a LED light bulb (which draws only 3-7 Watts), put in in a table lamp, plug it into your extension cord to provide adequate lighting for most needs. It is one of the best tips for the short term power failures I can think of, and the cost is minimal.
Cooking can be done outside on the grill (assuming you have a charcoal or gas stove, make sure you store extra fuel), and when your supplies run out, be prepared to start a fire for cooking.
Heating and cooling is a very difficult task to handle without a large generator, and is much longer subject for this post. I have some basic ideas for my needs including a small AC unit, fans for cooling, and a wood burning stove for the colder months. Please think about things you can do now to keep your family comfortable.
Please comment on any ideas you have that will help us all provide for our energy needs when things go dark.