Drink a gallon or more of water per day. This seems excessive, but high altitudes "thicken" your blood, which takes water from your system. You need to replenish this water by increasing the amount you take in. You don't have to do it all the time, but for the first week or so of high altitude living you should take in a lot of fluids.
Avoid caffeine and alcohol. These both dehydrate you and contribute to altitude sickness.
Take it easy for a few days. If you usually go for a five-mile run in the morning, take a few days off and go for a brisk walk instead. Your body needs to adjust naturally to the altitude and it needs time and energy to do so.
Ease into the altitude if you can. If, for example, you are going to Denver, which is a mile up from sea level, you could try to spread your trip out and stop on the way. This is not possible if you are flying, but if you are on a road trip a gentle easing into altitude is an excellent way to slowly adjust.
