Along the Icefields Parkway there are many side routes leading to hikes such as overviews and waterfalls. Although the Icefields Parkway could probably be driven in an easy day, you will get a lot more out of it, and see tons more, if you take a couple of days.
After Whistler on your way to Cache Creek (you can cut across to Kamloops from Cache Creek), there is the lovely town of Lillooette. Nearby is a provincial park with three glacier lakes, one above the other, each draining into the lower, from the glacier which feeds them, and each a brilliant turquoise colour. A great hike will take you to the top lake, and maybe see some wildlife along the way. I don't recall its name, it is almost two decades since I was there. Just that I was stunned by its beauty.
I would definitely recommend you take the spectacular Sea to Sky highway - 99, with a stop at Giribaldi Provincial Park, and the hike up to the waterfall there.
Edit: If you want to go to Jasper first, and then down to Banff, I would go via Prince George, and then take highway 16 - Yellowhead Highway - to Jasper. It is an incredibly scenic drive and there are several provincial parks along the way, as well as a number of small towns.
Do you have good maps? If not, it is not too late to order a Rand McNally road atlas from the RTA store. If you order it now you will have it in a couple of weeks. It will at least allow you to see the big picture, rather than being restricted to what fits on a screen.