Aspen Art Museum
This museum presents rotating exhibits often showcasing contemporary works of art from well known local and national artists. From this spot you will begin to find tributary trails and roads from which you can access downtown Aspen with its shops, galleries, and restaurants. You can spend lots of money and a little time here or vice versa. Enjoy a gelato and browse through one of the artist's workshops. Some of the streets require you to dismount and walk your bike for safety's purpose. The wide array of cruiser bikes w/ custom leather seats will fascinate those of you who are bicycle aficionados.
Like I said in the introduction, just because you're in Aspen doesn't mean you have to spend a lot of money. There are many sights to see and snacks to enjoy that are free or inexpensive.
For under $10, you should be able to catch a bus back to Glenwood Springs if you'd prefer not to ride your bike back. The transit station is located at Ruby Park near the base of Aspen Mountain on Durant and Mill Streets. Since the Art Museum is located on Mill Street, it's not too difficult to get to the transit station via the road or trail.