Destination Expert
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And, since I'm in Calgary, I can say that for your first visit to the Rockies with a fairly limited time frame, you might be better off skipping Calgary and focussing on the mountains. Calgary's a great city, but the Rockies are awesome.
I like Cbaarch's suggestion for skirting Calgary. i would even take it one step further: if time permits, take the Hwy 22 (Cowboy Trail) all the way from Pincher Creek to Longview, and then head west to connect with the Hwy 40 (Kanakaskis Trail) through Alberta's Kananaskis Country recreation area. Hwy 22 to Longview takes you parallel to the mountain front and through some beautiful ranchland (I believe famed Canadian country music icon Ian Tyson ranches near Lonview), between the Porcupine Hills and Chain Lakes Provincial Park, and past the historic Bar U Ranch (a National Historic Site).
Kananaskis Country, being further west than Hwy 22, takes you through the mountains to the southwest of Banff National Park. Note that there is no access to Hwy 40 from Hwy 22 from Dec 15 to June 15 each year. However, if you are visiting after June 15, you will be able to cross over the Highwood Pass, Canada's highest driveable mountain pass.
Here's that whole route on Google Maps: http://g.co/maps/cmd89
Of course, it's a huge (but very scenic) deviation, and will add travel time. Cbaarch's suggestion is the best from a time perspective, and mine is the worst. You could also consider just taking the Hwy 22 portion of the route: http://g.co/maps/8rjrf
You can even join up with the Hwy 22 later, by taking the Hwy 7 south of Okotoks (and past the Big Rock, a huge glacial erratic that was originally in Jasper: http://history.alberta.ca/historicsites/) to go west to connect with Hwy 22 northbound at Black Diamond - still very pretty country: http://g.co/maps/hwbzx