You can't take the park shuttle bus to trailheads east of Zion Canyon, although the long-distance East Rim Trail, which starts just inside the park's eastern boundary, joins trails that lead down into Zion Canyon's Weeping Rock Trailhead [1].
The fun Canyon Overlook Trail starts on the road east of Zion Canyon and features great views from the heights without the stiff climbs found on most other Zion trails. Allow about one hour for the one-mile round-trip; the elevation gain is 163 feet.
A booklet available at the start or at the Zion Canyon Visitor Center [2] describes the geology, the plantlife, and clues to the presence of wildlife.
The trail winds in and out along the ledges of Pine Creek Canyon, which opens into a great valley. Panoramas at trail's end take in lower Zion Canyon in the distance. A sign at the viewpoint identifies Bridge Mountain, Streaked Wall, East Temple, and other features.
The Great Arch of Zion—termed a "blind arch" because it's open on only one side—lies below; the arch is 580 feet high, 720 feet long, and 90 feet deep.
The Canyon Overlook Trail begins across from the parking area just east of the longer (west) tunnel on the Zion-Mount Carmel Highway.
Links:
[1] http://www.moon.com/destinations/utah/zion-and-bryce/zion-national-park/hiking/zion-canyon-hikes
[2] http://www.moon.com/destinations/utah/zion-and-bryce/zion-national-park/sights/zion-canyon