I'm from Calgary, but living in England at the moment!
If you're in the South, Fish Creek is HUGE and from what I understand you can go all the way into the city just through the park. I haven't done it myself but my mom and stepdad rode their bikes into the city from near South Centre.
You can walk along the Sheep River in Okotoks if you fancy a day trip out. Okotoks is beautiful, I worked in a little shop there on the old main road (Elizabeth St), and it's only about 25 minutes outside Calgary past Shawnessy.
Prince's Island Park right downtown is gorgeous too, especially on a nice day! I like the contrast of the park and the skyscrapers there. And don't forget the zoo, it's pretty big, you could walk around that a few times and see the animals at the same time.
As ilovescience said, Kananaskis, Banff, Canmore etc have some nice walks and nice places to walk around in the town as well.
There's several paths that seem to go past the Deerfoot. I don't know where these start or finish, to be honest, but I always see people walking their dogs there. And the Bow River is a beautiful place for a walk too.
I love Calgary, the people there are so friendly and I miss that a lot when I'm over here (not that English people aren't friendly, but they kind of need coaxing into it sometimes, whereas so many people in Calgary are just openly friendly and smile when you walk past, etc). I'm sure you'll be able to find plenty of places to walk! I hope you like your new home!