The biggest key for altitude compensation is water - water - water. Typical signs of altitude sickness include headache, nausea, breathlessness and fatigue. Much of that is exacerbated by the dry air in both the airplane to Colorado and the dry mountain air. Drinking lots of water (no caffeine or alcohol) will help you acclimate by countering the effects of dehydration.
There are also prescription drugs you can take before you go out that will help you acclimate. You need to see your doctor a few weeks before your trip.
When you get out to Colorado, don't try to push yourself too much. Also monitor your symptoms carefully. A mild headache and breathlessness are annoying, but normal, side effects. Don't be surprised if you can't ski nearly as much out West as here. If the headaches get severe (i.e., screaming pain), you get disoriented or feel like passing out, you need to get to a lower altitude quickly.
Just take it easy, drink lots of water and you should have a fine time. Aspirin may help with mild headaches.
I've been on several ski trips to Colorado in recent years (and lived in Colorado for 13 years), and on some trips, I had a lot of problems with the elevation, while on other trips, I didn't have any problems (outside of getting winded after walking up a flight of steps the first day or two!) There's no way to tell if you'll have trouble with it - sometimes you do, sometimes you don't. And, even if you have no physical exertion - and are in couch potato mode the entire trip - you're still at risk. Physical exercise doesn't cause altitude sickness, although it can exacerbate it.
Most people get by with few problems. A small percentage of people can have a serious reaction to altitude sickness, and it can be life threatening. The remedy is usually simple: get to a lower elevation. Ski patrollers out west as well as medical clinics are quite familiar with altitude sickness, so help is always close by. Again, very few people have the more advanced symptoms, but if they do, it needs to be taken seriously.
Drinking *tons* of water is great advice. (If you only drink when you're thirsty, you're not drinking enough!) Drink early and often! This applies to skiing, too. It's easy to get caught up in the skiing and not take breaks to rehydrate. I usually have a backpack with some bottled water, but most resorts have drinking fountains scattered about. As others mentioned, the air is very dry in Colorado, and you lose a lot of water as sweat when you ski (even if it is below freezing!)
Try and avoid alcohol, especially early on in the trip. Alcohol will greatly amplify the effects of altitude sickness. Avoiding caffeine is also wise since it's a diuretic and will compete with your efforts to stay hydrated.
One trick for adjusting to the altitude: fly into Denver and spend the first night there. You can find some cheap hotels near the airport (certainly cheaper than a night at a slopeside condo!) and head up I-70 the next day. Breck and those areas are only 2-2.5 hours west, and I-70 is usually in good shape. Denver is the "mile high" city - several thousand feet below the base of the ski areas. Staying there the first night will help your body start to adjust to the higher elevation, without suddenly being dropped at 9,000 feet.
(As a side note, another money-saving tip: if you rent a car directly from the airport, you will be charged all kinds of hidden airport surcharges. If you rent outside of the airport, you'll pay a lot less, I'm told.)
Try to take it easy the first day. I know it's tough, but sleep in - and don't feel like you need to make runs right up to closing time.
As Jim said, a mild headache and shortness of breath are to be expected and aren't cause for concern, although they're signs that perhaps you should take it easy.
You'll start to adjust a bit after a couple days, but in reality, it takes around 2-3 weeks for the body to fully acclimatize to a higher elevation, and then your lungs may only reach 90% capacity of what they could do at lower elevations. So even after a few days, don't think for a moment that you can keep up with the locals!
In the late '80s, I was involved in getting some people into Bolivia on Government business. The airport is at 13,500 feet and the city is at 11,500 - 12,000 feet. So high that some of the better hotels have O2 plug-ins on the room walls just like hositals so you can get a swig of O2. The folks who were sedentary and could fit into the "couch potato" definition had a really hard time, some having to be returned to sea level elevation. Many of these people were under 35, it just happened they didn't exercise regularly, were obese, or they smoked. For those who exercised regularly, the effects ranged from light symptoms to almost none at all. A couple of folks even went jogging the next day, and one of them went on a five-mile run with the House Hash Harriers (and many beers) a couple of days later.
I also spent three years in Bogota, at almost 9,000 feet. In the midst of training for the Marine Corps Marathon, I felt a little initial discomfort but I was running five miles a day within three days after arrival, and a 20 mile run within a week. That same weekend we hiked the mountains overlooking Bogota (13,500 - 14,000) with no ill effects. And that became my weekend past-time during my three years there. Again, on a trip to Quito (elevation 9,500), we went to climb the Cotopaxi the next day, making it to the 17,000 level. Being in shape works.
What I'm trying to say is that one should be in good shape already to be able to minimize the effects of hypoxia. If you go there with lots of O2 in your blood, the effects are minimal.
Altitude sickness can be equated with being drunk inasfar as both are the result of oxygen starvation. Altitude-induced hypoxia (hypoxic hypoxia) is the type of oxygen deprivation that concerns pilots and also the one that initially affects the person who arrives at high altitude. If in addition to this high altitude the person is of a sedentary nature, which will naturally pre-dispose the person to having less hemoglobin than an active person, then another type of aggravating hypoxia can be added to the equation, hypemic hypoxia. Even worse if the person smokes. And to that, stagnant hypoxia can aggravate the condition if the person has poor circulation caused by inactivity, obesity, diabetes, etc. To top it off, heavy drinking, smoking, chewing tobacco and many medicines will prevent the tissues from absorbing O2, normally called Hystotoxic Hypoxia. Best bet: Be in good physical shape, don't smoke, stay away from alcohol for a while, sleep, and over-hydrate.
There's an additional point to consider in high altitude skiing. Bring lots and lots of sun screen and lip balm. Above 5,000 feet, the level of UV radiation increases exponentially and dangerously. Even a short exposure to the sun can cause sunburn in exposed skin. Racoon eyes are quite common in those altitudes. One of the saddest cases I saw in my assignment in South America was a newly arrived US worker who arrived in Bogota on a beautiful sunny day and promptly proceeded to sunbathe on her deck. Her second day in Bogota was spent in a hospital with second degree burns all over her back.