Public transportation is available (Lynx, at
http://www.golynx.com) from the Transportation & Ticket Center to most points around Orlando and as far as Kissimmee and Sanford. A change of buses is often required. Lynx does not go anywhere near Daytona or Canaveral, however.
I'm not aware of a public transportation option to these destinations (though there clearly should be!), unless Greyhound makes the trip, and I'm not sure about that. Trains operate through Orlando and Kissimmee to Jacksonville, West Palm Beach and Miami, and Tampa - but not to Cape Canaveral or Daytona Beach. Seems like there might be a reserved 'charter' bus operation (Mears, or similar) to the Kennedy Space Center, but I'm not certain. If Greyhound does make any of these trips, the Kissimmee station is more convenient to Disney and can be reached by Lynx with a single change of buses (at Osceola Square Mall).