We're up here in New Hampshire and getting dumped on is a real problem basically every year. While the snow is bad, we worry even more about ice freezing on top of the already weighted-down branches, roofs, power lines, etc. In my mind, the biggest issue with losing power here is that, while a few of us have wood burning stoves or backup generators for emergency heat, many more of us rely exclusively on electricity for our heat and don't really have any financially feasible backup plans in place. We're a tough bunch who can get by well enough for a few days or more without the local grocery stores being open, without water, the roads being closed, or simply being snowed in, but freezing in our boots is a pretty bad scenario. Last year taught me an awful lot about the importance of staying warm when it's cold out, from personally having run out of heating fuel halfway through the winter and not being able to afford to buy more. Not fun!
Simple things that I feel help tremendously are, first and foremost, staying dry and insulating your area as much as possible. If you have a small room with no or few windows, insulating it as best as you can and keeping the door closed will help your body heat do its best to keep the temperature livable, if only in a section of your home. Don't use candles, open flame of any sort, or your oven to try to heat your home with. (Not only is it a fire hazard but it can be a danger because of carbon monoxide.)
Secondly, wear multiple layers of clothing. The air space between layers is a great insulator and helps keep you warm.
Thirdly, try not to eat or drink anything that is very cold, as the cold will lower your core temperature and make it more difficult to keep your body heat stable. Make sure you are not wearing too many layers, or wearing layers too tightly. If you start to sweat, your body temperature can drop drastically due to the amazingly effective ability your body moisture has at conducting heat away from your body. If you start feeling the beginning of sweat forming, take a layer of clothing off. More people freeze after sweating than do from being slightly chillier while bodily dry. Take it slow and easy.
Whenever possible, wear a hat and scarf, even while inside, and especially while sleeping, as we lose a lot of our body heat through our heads. If you have a lot of metal facial or ear piercings, remove them. They will freeze inside your skin and make it more likely you will get frostbite on those places. The metal will also conduct heat away from your body just in general.
If the air outside is very cold, there may be a danger of your pipes freezing. A few ways this can be helped are by putting pipe insulation on them, by using heating tape, or by letting your faucets "drip". If the heating tape is electric, however, and there is no power, it won't be very helpful. Letting your faucets drip is also a waste of water, and can really add up if you do it continuously, in your water bill. However, continuously running water can melt ice as effectively as it can defrost a frozen solid steak. It will also help relieve pressure from the expansion of the water when it freezes in a pipe. In my opinion, if it keeps your pipes from bursting, it's absolutely worth trying. When the temperatures get way below freezing here, we let the faucets drip, period. There are basically no downsides to pipe insulation, except that you will have to put the effort into fitting them correctly on the pipes. It is also important, if you can, to make sure condensation does not form and remain on your pipes, as this can also be a problem in conjunction with below freezing temperatures. This happens most often in our bathroom, after we take showers.
Beyond that, there are a few common enough tricks I think are fantastic in regards to warming up in the winter. One of my favourites is to throw a few handfuls of rice into an old sock and microwave it for a minute, then either use it as a handwarmer, footwarmer, neck warmer, or toss it under the covers to ease the frozen sheets. I have heard of people using a brick heated in the oven or fireplace, and putting it at the foot of the bed to keep them warmed up at night. Commercially sold chemical handwarmers are good in the same way, and can help un-stiffen fingers when the temperatures are too cold, and most importantly don't require microwaving or electricity for heating.
In regards to fabric, fleece, cotton, and wool are best for insulation and keeping temperatures steady. Feathers/down material/blankets/jackets/etc are just wonderful for keeping warm. Silk and polyester, in my experience, are terrible at keeping heat in. Newspaper is awesome as an emergency insulator, you can sleep with it right on top of you, or sandwiched in between blankets, or stuffed inside a jacket. It might crinkle during the night, but if it lets you stay warm, it's well worth sacrificing last night's newspaper for.
I'd be extremely interested to hear other tricks people use to stay warm and try to keep their homes warm, when there is no power to be had. In my case, budget is #1, and I imagine during "this economy" a lot of people are feeling the financial crunch the same as I.
"It is better to light a single candle than to curse the darkness."
"Knowledge is power. Power corrupts. Study hard, be evil."
"When I have a little money, I buy books. If any is left over, I buy food and clothes."