Welshmike:
Are you staying in Calgary the entire time? I'd definitely hook up with a member here and give the Bow a go. I wouldn't be a good choice as I am in Edmonton (3 hr drive, 2:30 when I am in a hurry) and there are dozens of people here with 10x the "know-how" and "where-to" when it comes to the Bow river.
You mentioned your trophy days are over, but the Bow has easy access and a shot at big fish. It is not the kindest river to newbies, so do go with someone.
Ok, so besides that, lots of good advice given here already. Literally hundred's of km's of streams to fish, NW, W, and SW of Calgary. Many of them do not require long hikes.
One variable you did not mention was solitude; do you care how crowded the waters are? The Bow obviously sees crowds, I'd also say the Oldman drainage sees the 2nd most people. The main stem of the Oldman is a popular spot, as is the Crow. I try to avoid weekends on those 2 rivers, though in last decade or so, fishing during the weekday is no guarantee either. Are you a friendly chap? Seems so, I ask that with this in mind; you can get away from people by getting to rivers via people's private land / farms / ranchers. Just need to ask permission. Many are friendly to fisherman, some are not. Tough part is finding who owns what parcel of land, and where is the damn farmhouse? But there are apps for that...
So I am just saying if there is at least a portion of the trip that you want some solitude, your best bet is smaller waters and creeks. Again, you don't need a 2-3 hour hike to get away from crowds, but at the same time, lots of bridge crossings may require 30 minute walks before you start to get away from people. My own personal rule on smaller water is if I see more than 1 vehicle at a creek crossing, I go elsewhere.
Also, (almost) every stream has public access off a bridge, and even if the stretch you are fishing crosses into private land, as long as you stay below the natural high water mark, you can keep fishing. Also, as per the earlier advice, generally speaking, the closer you get to the mountains proper, the more likely it is you are on crown land (though still be aware on the Oldman, Castle, and Crowsnest rivers).
If you want to have a lovely drive that is much less fishing oriented, but considered a classic "must do", is to drive from Calgary to Banff to Jasper then back (or do the full loop and go back through Edmonton and Red Deer). If you do end up in Jasper, pm the fellow who replied earlier here by the name of "Creekside". See if he can take you for a day of fishing on Maligne Lake. Stunning. Gorgeous lake. On the right day, some terrific fishing as well. Theoretically, if you left early enough, you could do Calgary to Jasper and back in one day (but it's pure driving and sightseeing, no proper fishing time), but its' 10x better as an overnighter or even 2 to 3 nights. Depends if you get sidetracked sight-seeing and/or some fishing. The fishing can be good along that drive, but is also VERY hit and miss; you need to know where to go inside the national parks. You would also need a separate license.
Lastly, more touristy stuff; if you like paleontology, go to Drumheller and see the Tyrell museum. World class natural history and dinosaurs. You could even fish the Red Deer river for goldeye and walleye.
Welcome to Alberta when you get here! Our province offers alot of variety. Have fun.
Smitty
P.S. If you plan to fish somewhere central, like near Rocky Mountain House or Nordegg (the Ram river system), pm me. I may be able to go with you.