Banff and Jasper National Park are beautiful, but be prepared for hundreds of tourists on all the popular trails around the time of year that you're considering - particularly around Lake Louise.
I'd consider spending part of your trip in Kananaskis Country (adjacent to Banff National Park). You can camp at the Kananaskis Lakes campground, where the scenery is every bit as spectacular as anything you'll see in Banff. I'd probably plan to spend 3 days in Kananaskis Country, 3 days in the Lake Louise area, and 3 days in Jasper National Park, around the ice-fields, if this were my trip.
Some day-hikes I'd recommend in the Kananaskis area would include:
Burstall Pass
Birdwood Lakes
Chester Lake + elephant rocks(although this one can be pretty busy in the Summer)
I'd recommend Mount Sarrail, Mount Northover, Mount Chester, Mount Smutts (difficult), Fisher Peak (difficult) maybe Mount Sparrowhawk (great view) if you enjoy scrambling to the top of mountains.
Additionally, the best way to see K-Country is by backpacking. I'd highly, highly recommend the Northover Ridge to Three Isle Lake backpacking trip. You can also do a backpacking trip out to Mount Assiniboine, which for my money, is probably the most beautiful place I've ever been to, but a bit of a slog to get there. There's a lodge that you can camp at, right beside the mountain if you don't have much back-country camping gear (or don't want to lug it 30 km to the camping location). You have to book early though.
If you're in Banff National Park, I'd check out the Skoki Valley, especially if you're able to do some back-packing. In the Lake Louise area, I'd climb up to the summit of Mount Temple. It's one of the highest peaks in the Canadian Rockies and the only one of its height that is easily accessible without any kind of climbing gear.