Banff is a beautiful town in the Rockies, about an hour and a half drive from Calgary. You can visit the hot springs, take a gondola to the top of the mountain, hike, and explore shops in town. Keep in mind you will have to purchase a park pass to enter the national park.
Lake Louise is another town near to Banff. The lake itself, in front of the Fairmont hotel, is incredibly picturesque (albeit crowded). The town is also in the national park.
If you want to explore further east, Drumheller is in the badlands of Alberta. The area is known for dinosaur fossils and has a very good archaeology and dinosaur museum.
If you have time and want to explore Canada, we also recommend driving or flying to Vancouver. Let us know if you need recommendations!
For guests with a car, we highly recommend a day trip to the Leighton Center, about 45 minutes south of Calgary. It's the house of painter A.C. Leighton, who moved to Canada to paint posters for railway companies, and is full of beautiful art. It's also, in Carmen's opinion, the best view in Alberta.
A little closer than Banff, the Kananaskis area is the start of the mountains. There's a ton of good hiking here - ask if you want recommendations on trails!