My top pick for 4 days would definitely be to see the Rockies.
Plan A: Edmonton-Jasper-Lake Louise-Banff-Calgary. (And drop the rental car & take Red Arrow back to Edmonton from Calgary ;-)
The Icefields Parkway between Jasper and Lake Louise is a "must",provided the weather cooperates and there is enough time in your schedule.
Plan B would be Edmonton-Jasper-Lake Louise-Banff-Lake Louise-Jasper-Edmonton. (Yes, I am suggesting that you skip Calgary.)
The mountains are definitely a beautiful part of Alberta, but are only a part of the province. I would head from Edmonton to Drumheller (Try to drive Hwy. 56 instead of Hwy 2 for scenery, although it adds a few hours.), then head through Calgary to Banff and up to Jasper. If you have to choose one access point for the mountains, I prefer Jasper as it's a little bit less touristy and I've seen more wildlife there. However, Banff is closer to Calgary than Jasper is to Edmonton so it depends how much time you want to spend in a car. If you can do the loop, you get to see both.
Are your 4 days in Edmonton set in stone with a family visit or conference? Because with only 4 other days to spend in beautiful Alberta I'd definitely choose the mountain scenery too. Drumheller is interesting and with a full week, more doable.
I've lived in Jasper, Edmonton, Red Deer and Calgary and would also recommend the Edmonton/Jasper/Lake Louise/Banff route with the option to fly out of either Edmonton or Calgary. The Icefields Parkway is unforgettable!
If you decide you want to see a little more of the province and are flying out of Edmonton, then I might take the 6 hr drive from Banff back to Edmonton via Cochrane, Caroline, Sylvan Lake and onto Edmonton.....http://g.co/maps/k6m44
You could do 2 nights in Jasper and 2 nights in Banff.
Have a look at this post and video to see some of what you can look forward to. :)
I agree with albertaloner. Drumheller is a must, especially if you have kids. The dinosaur museum (Royal Tyrell) is world class. But, of course, you don't have to have kids to enjoy the badlands and dinosaur country.
HIghway 21 is a great less travelled road, and gets you to Drumheller faster. It is also close to my fav buffalo jump, Dry Island buffalo jump. Gorgeous area.
You would not go through the little towns that Tobie mentioned, but to me it's worth it to cut an hour off the car ride which would be better spent in the mountains.
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