Aspen itself is famous for its fantastic ski slopes, and a week or two spent here will have plenty of variation to keep you occupied. A few miles away is Snowmass Village, for a slightly quieter, off-the-beaten-track skiing experience.
If you have time, be sure to take a drive around some of the gorgeous local towns, which include Glenwood Springs. Aptly named, it features natural hot springs to spend an afternoon relaxing after a few vigorous days of skiing. The Grand Mesa National Forest is also well worth a visit for photography enthusiasts or those who simply enjoy amazing vistas.
Denver is a few hours' drive from Aspen, and is Colorado's main city. Truly metropolitan, it's well worth a visit for its fantastic range of museums and landmarks, including the Chamberlin Observatory and more unusual finds like the Denver Firefighters Museum.