There are a lot of open spaces within the city of Calgary. The largest parks would be Nose Hill park in the northwest and Fish Creek Park which crosses most of south Calgary.
Fish Creek has miles and miles of bike and walking paths as well as that great man made swimming hole Sikome Lake. I personally spend many hours in the summer time walking and biking this great park. I like to go biking in the evening around 8 pm when it is still warm enough and you will see a lot of wildlife such as deer, eagles, hawks, and coyotes. Much of the trail system follows the bow river, so there are many places to stop and take in nature while catching your breath. If you are a fisherman, you can partake in world class fly fishing although you probably wouldn't want to eat the fish due to pollution of the city that pours into the river from the city's street drainage system.
Here is a nice video that shows you what Fish Creek Park has to offer.