After you enter Yellowstone National Park from the West, you will come to Madison Junction. Madison Junction is where the Gibbon River and the Firehole River merge to become the Madison River. Just up the Firehole River (south) from this merger is the Firehole Canyon Drive, which is a one way road you can drive south on.
While these two rivers may seem like they should be similar, the Gibbon and Firehole are very different in at least one way. The Firehole River is very warm. That is because the Firehole River travels north from the Geyser Basin and many of the geysers runoff fills the river. Because of the warm water, there are several swimming holes visitors can enjoy.
The Firehole Canyon Drive follows the Firehole River on its eastern side. Looking over the gorge and the cascading waterfalls and rapids, you gaze upon National Park Mountain, 800 feet above you.
If you are just driving through quickly, (and there are good reasons to want to get further south into the geyser basin) just make one stop at the Firehole Falls. It is by far the best spot on the drive.
If you have a lot of time, make it a point to bring a swimsuit. There is a very cool stop that is clearly marked for swimming. If I remember correctly, it is toward the end of the drive on toward the South end. This is a deep pool and the water temperature was about 75 degrees when we visited. You will know you are there because you can walk down to it and you will be bombarded with signs saying don’t jump off the cliff’s, etc….
Just after the Firehole Canyon Drive, the river flattens out and there are a bunch of picnic sites scattered through it. By the way, fly fishing is permitted on this river. Check with a visitors center for more information.