Green Canyon is the best and is gorgeous. It is why the Logan area is one of my favorite spots in the world.
This place is gorgeous in the summer. It lives up to its name. We often go camping there. You can also head to town to get supplies. It is close to town. We like to spend one to two days camping.
The winter is great too. There is a huge cave. You hike up a mountain. The view from it is beautiful. You can see a long ways out. There is also a smaller cave a ways down. One year we were allowed to stay at a yurt. It was so much fun.
The canyon is not a far drive. You can drive a ways in if you want to, and hike only as far as you want to.