Some of the answers here are, frankly, dangerous.
Aerobic training will not prepare you for altitude
Aerobic training such as walking up inclines is a given if you want to enjoy a challenging trip, but will not prepare you adequately for altitude work. I've climbed with highly conditioned endurance athletes, and can tell you from experience that they have much the same risk of altitude sickness as everybody else.
Altitude sickness is a significant risk at 14,000 feet (4,200m)
At 14,000' (4,200m) you're only getting 60% of the oxygen you're used to at sea level. Around 80% of people are prone to altitude sickness of varying degrees if they ascend quickly to this altitude. Are you feeling lucky?
Without acclimatisation, most people will experience mild altitude sickness from around 8,000' (2,400m) with a significant risk of disabling sickness from around 12,000' (3,600m).
Don't underestimate the dangers. Even mild sickness is extremely unpleasant and can affect your ability to move safely on technical ground. Severe symptoms can be pretty much paralysing.
Bottom line: it would be extremely rash to go straight up to 14,000' (4200m) without acclimatising properly - you could end up in severe difficulties.
It is essential to acclimatise properly before ascending to 14,000 feet (4,200m)
If you want to climb fourteen-thousanders, you only have two proven alternatives - either you allocate 3-4 days to training at lower altitude, or you pre-acclimatise with altitude tents and masks.
I only have direct experience of traditional training. I'm prone to sickness, so personally I find it best to be conservative and climb up to around 9,000' (2,700m) on the first day while sleeping low. Even this can be quite uncomfortable. Then I'll add 1,000' (300m) or so a day if things go OK. If I'm relatively symptom-free on an 11,000' training climb, I'd feel ready for 14,000' (4,200m).
This regime is for typical alpine work where you are sleeping low. If you are on a traverse or trek where you will be sleeping high for multiple nights you have to go slower and bail out at the first sign of symptoms. Here is a sobering account of a fit young woman who died of edema after ascending to 14,500' on the 5th day of a trek in Nepal.
The only proven alternative is to pre-acclimatise using altitude tents and masks. This is quite an expensive and disruptive process, but may be worth it if you simply can't find the time to acclimatise on the hill. The equipment can be hired or purchased if you don't have a suitable facility nearby.
Medications are for extreme circumstances. They have significant side-effects and are strongly discouraged for normal recreational use.
