A fun hike or mountain bike trail is the Rim Trail. Even though there are two places to hop on the trail, I personally recommend starting at the side near the Snowmass Mall off Divide Road.
The trail is a series of tame switchbacks that traverses up the hill through brush and trees. You'll know you are at the top when you hit the glossy yin and yang sign made of stone that is a permanent fixture of the mountain. It is very fitting, because when you stand on the yin and yang sign the view you are looking upon is a 360 degree panorama of Snowmass and the surrounding area. To the north are the mountains of Vail, Mt. Daly is to the south, Mt. Sopris to the east, and west lies Independence Pass in Aspen. It is one of the few places where you get this type of all-encompassing view.
At that point you can go back down the way you came or you can continue on the path that follows the rim of the hillside, hence the name, which spans the length of Snowmass Village. The trail is open in the summer for mountain biking and hiking. If you are hiking and chose to do the rim be weary because mountain bikers have a tendency to rip through the downhill sections. In the winter it is an amazing hiking or snowshoe trail although if you are the first to venture out after a snow storm you might be blazing the trail for others because it is not commonly used by many in the winter.