This looped trail climbs the west side of the canyon for good views of several named features of the park, including the Three Patriarchs and the Streaked Wall. (During the main summer travel season, horseback rides are given on this trail, so you might run into churned up soil, dust and other horse souvenirs.) The trailhead is at the footbridge over the Virgin River.
This is absolutely the best map available for hiking Zion National Park. It lays out the whole park three-dimensionally and shows more than 40 trails. You'll easily be able to see how to reach the trailhead and where the trail runs.
During the main season, you'll need to take the shuttle. Off season: From the visitor's center at the south entrance to Zion National Park, proceed north along the main scenic drive until you reach the Court of the Patriarchs pullout. A small parking area is on the left. The trail begins across the footbridge over the Virgin River.