This may be out of your "day's drive" from Reno, but two years ago, my family was able to visit Yellowstone, which according to Google Maps, is a 10 hour drive from Reno. We only spent 3 days driving around YNP, because we also visited Glacier NP in the same trip. We flew and rented a van, but if people had more time and no problems with driving distances, a road trip would work ok.
We only did a small hike in Yellowstone, along the Yellowstone River. But there are some epic day hikes in Glacier. Unfortunately, Glacier is more than that one-day distance.
