Water: your second highest priority
After dealing with medical emergencies, the priority is your water supply. You can live without food, but each person needs a minimum of around a half a gallon per day, with a gallon per person per day being recommended.
If you'll be doing hard work, or you're in a hot and/or dry environment, more than a gallon might be necessary. Note that some mountains are generally dry areas, and if you'll be working hard in the middle of the day in a desert area in the summer, you'll probably need at least two or three gallons per day.
The easiest way to get your requirements is to store it: at least three days' worth per person at home and in your car. Store it in clean, tightly sealed plastic containers. Other types of containers are possible, but bear in mind that glass tends to break. You can buy already-filled bottles, or just buy the bottles and fill them yourself.
For the latter, you can purchase large Sparkletts-style bottles at wholesale supply stores. For smaller quantities, some soda pop is also available in 3-liter bottles. I have extensive experience using Shasta 3-liter bottles (as added weight during training hikes), and they're very tough. In any case, use a bottle brush to clean the containers first and rinse them well before filling with clean water.
You will probably want to replace the water supply every six months either by purchasing new filled bottles or refilling the ones you already have.
You will probably want to avoid the flimsy gallon and 2.5 gallon cloudy plastic containers, since they might break open if you have to carry them. Water weighs 8 pounds per gallon, so even if you're an experienced backpacker in very good shape and you have a large backpack you won't be able to carry much on your back.
If you've got access to water, but it's not clean drinking water, or you don't know whether it's safe, you'll need to purify it. Don't do this with runoff storm water or something that might contain harmful chemicals or is excessively polluted. Find another supply. If it's murky or contains other foreign matter, filter it first with cheesecloth or similar.
Three methods of purification are described in this PDF file from the Red Cross. The first is boiling. Bear in mind that water boils at a lower temperature at higher altitudes. So, you may need to let it boil longer if you're in the mountains.
The second technique uses bleach, and the third involves creating a bit of a still. In the distillation technique, you need to make sure that the lid's handle isn't made of something that's going to break down or even melt when exposed to hot water, and likewise with whatever you use to attach the cup to the lid. They also suggest allowing 20 minutes, and that assumes that you have access to a gas supply. In the case of backpacking stoves, this is a highly inefficient technique which will leave you with little fuel for other uses.
There are also hand-driven water purifiers. There's some more information here, but the specific models mentioned are outdated.
If you have access to water from high mountain streams, you might be able to get away with using a hand-operated water filter. Note that this is less sure than water purification.
You can collect rain water, or let clean snow melt. Don't eat snow, because you'll expend a lot of energy melting it, which will make you require more food and water.
Various ways of distillation are described here and here, although the second is a bit involved, to say the least. There are good adhoc techniques herehere. There are plans for building a large distiller here.
In the case of the bird flu, check here.
If you're absolutely sure that they are safe to eat, you can also obtain water from plants: leaves, roots, and so on. In an absolute emergency, you may be able to place a large number of dew-moistened leaves in a plastic bag, close the bag tightly, and put it in the sun. That will form a bit of a still. The water would then need to be at least filtered.