We don't have cities right in our mountains like Vancouver. The upside of that is that our mountains don't have cities in them.
You're probably better off living in the west than the east, but expect it's about an hour's drive to the mountains from most of Calgary. 45min from NW Calgary to the mountainfront, and 30 mins from the deep south to the start of the Elbow Valley.
If you're working downtown five days a week, and heading out to the mountains two days a week, you're probably better off getting a place to live that is close by your work.
You could live in the west end of Calgary and commute in and still find it ease to get out on the weekend, but that will be a 30-60 minute commute every day for a 15-minute savings on the drive to the mountains on a weekend morning.
I would suggest getting a place downtown or nearby where you can walk or bike to work, depending on your preference and budget. That will still be very little extra drive on a dead Saturday morning heading to the mountains.
Anything right in downtown is awesome, but pricey. Kensington, Sunnyside, or 17ave S area will be a little cheaper with a bit of a longer walk. Hillhurst, SAIT area, just below 17th Ave S, will be very reasonably priced but a longish (30-50min) walk, or a nice short bike ride.
If you live further out, and want to bike, along the river or another straight corridor is nice for a simple bike ride. Living right on the train isn't too bad, as the train does come regularly in rush hour, but it is crowded, and does stop running in bad weather.
Here is a picture of the downtown Calgary area:
Downtown is circled in Black.
The green areas are a pretty short walk to downtown, though still a little pricey. The yellow areas are walkable, or very short bike rides, but a little cheaper. The orange is walkable, though a longer walk, or a short bike ride. SE of downtown is the not-so-nice part of the city, but Ramsay has always seemed nice enough to me.
I live in the orange circle second left from the top. I've a very slow mover, and it took me 15-20minutes to bike downtown, or 45 to walk, when I used to live there.
I always calculate my driving on 1 hour to Canmore as well.