Bozeman has many, many hiking trails from town up into the mountains all around the valley.

Here is a you tube video about one... I hiked to Hyalite when I was in college during the late winter when the snow was deep on the ground. A couple of friends and I camped out on the snow over the weekend and then hiked back down to the road.
This video is of a guy hiking in late Sept this year, when snow came a bit early.

A Hike Up to Hyalite