I have always wanted to hunt bear or moose but I figure I should start smaller with deer or elk.
Do you have your WIN card / hunter training course yet? Many species in Alberta require a draw, and the draw period for next season is in June. I suggest putting in a draw for antlerless mule deer and whitetailed deer. Hunting them is fun, your chances of harvesting one will be good, and they will help you get experience with field dressing, care of meat, and of course provide you with some fine eating.
If you were hoping to hunt around Calgary, you'll be waiting for several years before you can draw a moose tag, so start applying next year. Some WMUs for elk are general tags, others require a draw. There are many WMUs open to general whitetail tags, and black bears are also a general tag. If you haven't done this yet, stop by your local Canadian Tire and pick up a copy of the hunting regulations for 2008. They will give you an idea of what you can hunt with general tags, and what you will need to get drawn for.
I would offer to take you out sometime, but I don't think I have time for it this season. Here is an organization that can possibly help you out if you need to take your hunter ed course still:
http://www.aheia.com/index.cfm
I believe that they also have workshops and mentoring programs, so if you can’t find an experienced hunter to take you out, you might find someone there who can help get you into hunting. If you don't get out for big game this fall, you might also want to consider trying coyote hunting this winter. It will help get you into the hunting mindset.