Banff National Park - Of course it would be more ideal to pay a longer visit to the Canadian Rockies.  However, if you are pressed for time, you can see some of the highlights in a single day.  On a day trip you could fit in a visit to Moraine Lake (for the best view, walk to the top of the pile of rocks at the north end of the lake), Lake Louise and Banff townsite.

Royal Tyrrell Museum  - The writer and palaeontologist Stephen Jay Gould called the Royal Tyrrell Museum "the world's best dinosaur museum."

However, there's more to the Tyrrell than just dinosaurs.  There are lots of interactive games and displays, there's a conservatory featuring the modern descendants of dinosaur-era plants, there's an amazing exhibit featuring the bizarre Pre-Cambrian fauna first found in Canada's Burgess Shale, and there's a nice display on Ice Age animals such as sabre-toothed tigers (although by the time you reach this part of the museum, kids are often too tired to appreciate it). 

The museum is located in Drumheller, a town of 8,000 that is located about 145 km (88 miles) from Calgary. It takes between 1½ hours and 2 hours to drive there from Calgary.

The drive to Drumheller takes you east from Calgary, across the flat prairies. Some people would consider the drive to be boring.  However, if you are willing to see beauty in every landscape, you may be able to appreciate the "Big Skies" of the prairies.  The bright yellow of the canola fields is attractive in summer. 

Just outside of Drumheller, there are two canyons, Horseshoe Canyon & Horsethief Canyon, from which you can view the unusual "badlands" terrain carved out by the Red Deer River.  It can be interesting to walk through badlands, because people find that their sense of scale and distance is thrown out of kilter. There is also a short interpretive badlands walking trail just outside of the Tyrrell Museum, in Midlands Provincial Park.

During the winter months, the Tyrrell is open from 10 am - 5 pm and it's closed on Mondays. It's a treat to go through it during most of the year when it's quiet.

In the summer months the museum is open 7 days a week, from 9 am to 9 pm.  Unfortunately it is busy in July and August, and that does detract from the experience.  In the summer months, it's better to visit from Monday to Thursday. In July and August you’ll find that Fridays, Saturdays and Sundays are extremely busy. If you must visit on the busiest days, it's better to arrive right when the museum opens its doors at 9 am or else wait to go through between 6 pm and 9 pm.

People who have mobility problems may rent a wheelchair for C$2.

The cafeteria in the museum is adequate but not spectacular. If your timetable allows, it may be preferable to have lunch in the town of Drumheller rather than at the museum. However, if you are short of time, it’s all right to eat at the museum. 

If you do the museum as a day trip from Calgary, you should allow at least 7 hours for it (2 hours to drive there, a minimum of 2 hours to go through the museum, an hour for lunch, and 2 hours to drive back.). However, if you stop to look at Horseshoe Canyon on your way into Drumheller, and if you browse through the fossil stores in the town of Drumheller, the trip easily could take you upwards of 9 hours.

Some people like to end their daytrip to Drumheller with an evening at the Rosebud dinner theatre, a 30-minute drive from Drumheller.

In July, the Canadian Badlands Passion Play takes place in an outdoor amphiteatre.

When you've completed your visit to Drumheller, you may want to drive to Banff without getting caught up in Calgary's traffic.  To bypass Calgary, drive Drumheller - Beiseker - Airdrie - Cochrane - Canmore - Banff.  It's worth stopping in Cochrane for an ice cream from Mackay's.  That route takes 3 hours, not counting stops. 

The information about the Royal Tyrrell Museum and the town of Drumheller came from this thread on the TripAdvisor forum

Kananaskis Country - Do a circuit through the hamlets and small towns of Bragg Creek, Turner Valley, Black Diamond and Longview, the beautiful Highwood Pass, Upper and Lower Kananaskis Lakes.  Stop a couple of times to walk or hike in the vicinity of the Highwood Pass and Kananaskis Lakes.  After you leave Kananaskis Country, you can return to Calgary on the TransCanada Highway (Hwy #1).  If time permits, you can take a detour to the town of Cochrane where it is worth getting an ice cream from Mackay's before you return to Calgary.

Head-Smashed-In Buffalo Jump Interpretive Centre - Drive south of Calgary on Queen Elizabeth II Highway.  Just before Fort Macleod, turn west to Head-Smashed-In Buffalo Jump Interpretive Centre.  After going through this museum that depicts the traditional lifestyle of the First Nations people of the prairies, continue to the small town of Pincher Creek.  Then return to Calgary via the scenic Cowboy Trail (Hwy #22) that meanders through the foothills at the eastern edge of the Rocky Mountains.  Tip : The selection of food in the museum's cafeteria is poor.  You would be better off taking a picnic lunch with you (there are picnic tables just outside of the museum).  Alternatively, you could have lunch in Fort Macleod or Pincher Creek.  Caveat : This museum does not appeal to everyone.  It especially may not appeal to children who are of elementary school age or younger.  The exhibits at this museum are not "in your face" in the way that the dinosaur skeletons at the Tyrrell Museum are.  You have to read the descriptions next to the displays and use your imagination to visualize how life was for hunter-gatherers in this demanding natural environment.  However, you would find this museum fascinating if you were interested in different cultures.  It would be all the more compelling if you also had a chance to visit the Royal British Columbia Museum in Victoria or the Museum of Anthropology in Vancouver.  Then you would have an opportunity to compare the contrasting lifestyles to which two very different ecosystems gave rise.       

Edmonton - Alberta's capital city is three hours' drive north of Calgary.  An attraction that draws many visitors to Edmonton is West Edmonton Mall.  Edmonton also has other attractions, like Fort Edmonton Park, the Provincial Legislature, the Royal Alberta Museum, the Muttart Conservatory, the Ukranian Cultural Village, and Whyte Avenue.  Although it is possible to visit Edmonton during a day trip from Calgary, it is better to spend a night or two there.