We (friends and myself) have done (by car) Quebec, Montreal, Ottawa, Toronto (for me - quite dull for 100's of miles in a straight line - and pretty much totally flat), then onward to Thunder Bay (fascinating city, right on the banks of Lake Superior) and Winnipeg (not the most exciting place - but by far the most cultural and friendliest of all the cities we visited), where we took a weeks break. After that we did Regina, Medicine Hat, Calgary, Banff, Kamloops and Vancouver.
The most boring bit of the journey for us was through Saskatchewan. Just a mighty great big bit of lots of nothing.
On the way back we did Edmonton and Saskatoon, before dropping off the car in Winnipeg and getting the flight out of Canada. Edmonton wasn't at all interesting (apart from having the world's largest shopping mall - not interesting for me though). Saskatoon was dreadfully boring, but the people were so friendly.
Once you're past Saskatoon heading West, the hills soon become mountains and it all gets exceedingly pretty.
Loved Canada and it's people very much and what a huge country too, but travelling from East to West - there are days of nothing but flat and nobody until you're well past Western Manitoba.
Having never been to the east (besides the Toronto airport, which doesn't really count), I would still have to say go east. Honestly, I'm a Calgarian, and though you can find things to do here, you will have way more variety in Montreal/Toronto/Quebec City. Calgary livens up about once or twice a year: during the Calgary Exhibition and Stampede in July and during the National Hockey League play-offs- if the Flames make it, that is. Besides that, Calgary can be kind of dull. And Edmonton . . .yikes. I've got to say, a cure for insomnia.
The reason to go to Calgary in the first place is to see the Rockies, an hour's drive from Calgary, as others have said. But since you will not be driving, it will make it difficult to get from place to place. Well, you could, theoretically, hitchhike or take the Greyhound bus, but you'll need a good dose of patience if you want to do that, and more time. When driving fits your plans, I would definately recommend a trip to the Canadian Rockies, though. The beauty is unparalleled. I guarantee you will not be disappointed. Nothing compares. . ..
Have a great trip!
I did internships in both Calgary and Toronto and have to agree with most of the comments here that there is more to see in Toronto/Montreal/Quebec than Calgary/Edmonton. Edmonton has a really, really large mall (cinema, amusement park, water park, shops) and Calgary is small. I personally have no desire to see either city again (however, as another person posted, Banff and the Rockey Mountains are beautiful).