Green River Narrows Hike is located in the Green River Gamelands. This area is well loved by kayakers, but it also a great area for hiking. Most of this hike is pretty casual until you get to the last .3 miles which is straight downhill. There are ropes to aid in your descent during this section.
I wouldn't recommend bringing a dog unless they are comfortable off leash and have good recall. Trying to do this section with a dog on leash would prove very difficult. Once at the bottom you can enjoy the gorgeous rapids, and if it's nice enough out find a spot to swim.
I'd easily spend all day here having a picnic, reading a book, and watching the kayakers go by.