Getting a license is easy... just go to Canadian Tire (customer service desk), Wholesale Sports, etc., and buy one. I think non-resident Canadians pay the same as Alberta residents, so the cost will be just a tad over $25, plus you'll need to buy a WIN Card, which I can't recall what the cost is. You will receive a copy of the regulations when you buy your license, but they are also available on AO here:
http://www.albertaregulations.ca/fishingregs/
As for fishing, you can start right in the city on the Bow River, which is one of the best trout fishing rivers in Canada, if not the world. Anywhere from Fish Creek Park downstream to Caresland is good, fishy water. Do you fly fish? People come from all over the world to fly fish the Bow. Spin fishing is effective too.
I would suggest picking up a copy of a magazine called "Barry Mitchell's Guide to Fishing Alberta." It has directions to most fishable waters in Alberta and also information on what you can catch. If you don't mind a bit of a drive, try going SW of Calgary to Beauvais Lake or Beaver Mines Lake. Both are stocked with trout, and the scenery in the mountains is an added bonus.