The Happy Isles are on the Merced River. From the Yosemite shuttle bus, you exit at the Happy Isles stop to hike the Mist Trail (which leads to Vernal and Nevada Falls). So take a few minutes to visit these charming little islands in the river. Check out the Gaging Station. It contains a set of instruments that measure the water's temperature, speed, composition, and other characteristics.
The Merced River flows down through Little Yosemite Valley, down Nevada and then Vernal Falls, to Yosemite Creek. The Mist Trail follows this valley. It's not only the falls that are beautiful, but also the views along the way. Don't forget to stop every so often, look back down the valley, and admire the spectacular scenery.
If you spend any time in Yosemite Valley, you'll know this river. It runs right through the middle, and just upstream are Vernal Falls, and then Nevada Falls. My favorite areas are upstream of Nevada Falls, but that's a hike.
One of the scenic areas in the park are the views of the Merced River. It is one of the most photographed spots due to its serene location. Its locale can be found on a good map of the park.