I would strongly suggest hiking to the Lake Agnes Tea House above Lake Louis. Many people do head up there so it won't be in solitude but the views are awesome and it is a neat experince to have done a decent hike and then have some scones and tea. You can continue on past the tea house and up to a view point which adds a bit of veritcal but much of the crowd drops out at the tea house. If you have the time and inclination, you can make a loop hike and hit the Plain of Many Glaciers Tea House on the way out.
Make sure to try and get out of Banff the town. It is very busy and mostly nick nacks. However, just above Banff there is a via ferrata at Mount Norquay that was a really good time.
@Elena, we are headed up there this summer and here are a few things we are doing:
Mt. Assiniboine over 3 days from Sunshine to Mt. Shark.
Kayaking on Lake Magline in Jasper
Day hikes including Wilcox Pass, Bald Hills, and Bow Summit Hike. There seems to be a ton of fantastic day hikes in the area so you can't go wrong.
Johnston Canyon (take it all the way to the Ink Pots) is the most amazing hike we did in Banff. The tea houses are nice, but those trails are crowded. Most people don't hike beyond the falls in the canyon, so the crowds thin out significantly. My girls (7 and 11) did this and loved it. You climb up into a high meadow with the river and ink pots, and you're surrounded by these rocky peaks. It is beautiful!
The Vermillion Lakes are a fun side trip, too. They are easy to explore and not too crowded.
I second Lake Agnes Tea House, I really enjoyed eating there and the views from the trail were great. Just make sure you get there early enough, it does get crowded. Bow Glacier Falls Trail was a cool hike.
If you're able to I highly recommend going up to Jasper National Park. There's a lot less people and some cool hikes up there.