Here are some of the good trails, for hiking or cross-country skiing, that are accessible from the town of Banff:

Sulphur Mountin
From Cave & Basin to top of Sulphur Mountain (Sanson Peak)(hard)
Sulphur Mountain Loop
From Bow Falls up the Spray River on one side of Sulphur Mountain, and back down Sundance Creek to Vermillion Lakes on the other side of Sulphur Mountain (long)Family hiking
Tunnel Mountain Peak
From the Banff Center to the top of Tunnel Mountain (short & easy)
Hoodoos
From the Banff Centre to the Hoodoos (short & easy)
Mount Rundle Peak
From Bow Falls to the top of Mount Rundle (hard)
Stoney Squaw Mountain
From the top of Mount Norquary Road to the top of Stoney Squaw Mountain (hard)
Cacade Mountain
From the top of Mount Norquay Road up the back of Cascade Mountain (really hard)
Mount Rundle to Canmore
Along the south bank of the Bow and along the base of Mount Rundle to Canmore (by the Nordic Centre)
Lake Minnewanka
From just inside the Park Gates, up Carrot Creek around to the far side of Lake Minnewanka.
Healy Pass Trail (shown, above) From the Sunhine Village parking lot, this 22 km trail takes you to the Continental Divide with stunning views of Mount Assiniboine and of Pharaoh Lake

Other trails in the Rockies area