There's lots of great backpacking all across the continent, and lots of good suggestions above.

Grand Canyon is a once-in-a-lifetime experience, but you need to plan ahead and know what you're doing. We went in May some years ago: still some snow on the south rim, but at the bottom, we typically got up at first light, hit the trail around dawn, and by 10 am it was too hot to hike.

If you happen to make it to the Canadian Rockies, PM me and I can make some specific suggestions depending on time available and season.