Hello Lori!
I love using the Walt Disney World Transportation System to travel around "the World"! The Disney Skyliner has become my favorite method of gliding above the ground to get to Epcot and Disney's Hollywood Studios.
Typically, the Disney Skyliner starts operating 45 to 60 minutes before Epcotor Disney's Hollywood Studios opens on that particular day. Remember that park hours can vary day to day or during different seasons, so the Disney Skyliner start times may vary. At the Skyliner stations, there should be signs alerting Guests to when the Skyliner is operating on that day, according to my experiences. You can also ask a Cast Member at any Skyliner station or even at your Walt Disney World Resort, and they should be able to let you know the operating hours for that day.
I hope your Disney Skyliner flight is magical on your next Walt Disney World vacation!
Marc L.