From the inner city you can get on to Calgary's pathway system which has more than 500 kilometres of paved trails. There are various places where you can find single track off the pavement, including Fish Creek, which I think is the largest urban park in Canada, bigger than even Stanley Park.
There is also some single track in Bowmont Park, near Bowness, and 12 Mile Coulee. But my favourite XC riding in the city has always been in Nose Hill Park in the northwest. I lived right next to the park for a long time and rode literally thousands of kilometres there each year, sometimes every month of the year when the Chinooks melted the snow. Canada Olympic Park (local ski hill) has downhill biking with lift rides, but hopefully some day they will open the XC trails in a new area to the east of the ski hill that was purchased about five years ago. There is are some awesome trails there, which were used once for a World Cup race, and which I poached one day.
Unfortunately in recent years there has been a trend to restrict off-pavement riding in many areas, so some trails we used to ride in Fish Creek, Bowmont etc are now officially off limits (of course only to cyclists, not dog walkers).
I know you mentioned not having to drive to ride, but during our long summer daylight hours, many people throw the bike in the car after work and drive the half hour west of the city to the Elbow Valley and some of the best mountain biking anywhere.
Some links with maps that might be of interest:
www.tpr.alberta.ca/parks/fishcreek
www.tpr.alberta.ca/parks/kananaskis/pdf
www.braggcreek.ca/kananaskis/trails
www.calgary.ca/Pathways
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