Scout Lookout via West Rim Trail
If you're interested in getting close to Angels Landing without doing the scary part, this moderate out-and-back is for you.
Pay for your pass (https://www.nps.gov/zion/planyourvisit/fees.htm) at the park entrance just north of Springdale, Utah, then park at the visitor center and catch one of the frequent shuttles (https://www.nps.gov/zion/planyourvisit/shuttle-system.htm). Get off at The Grotto and cross the road for the trailhead.
There's no special permit required for this hike, but the trail gets crowded fast; catch an early bus if you can.