There are a lot of designated spots but just keep your eyes open and you'll find all sorts of exciting shots. My favorite in the Magic Kingdom is Cinderella's castle - if you walk to the left towards Tomorrowland, you'll see a walkway off to your left (rose garden) that goes down towards the moat. There's a pavillion at the foot of the walkway - about halfway down this path is an excellent spot for a morning/early afternoon shot. It's also a great spot for watching the fireworks at night!
My all-time favorite is the statue of Walt and Mickey at the end of Main Street and in front of the castle - Best time is when the park first opens. I've sprinted from the entry turnstiles to get the best shot - no one around or behind the statue - and the castle is the perfect backdrop.
In EPCOT, a shot of the fountain of nations with Spaceship Earth behind it is always beautiful (especially if you catch the water in flight).
Keep an eye out for the posted Kodak spots - they are good photo ops. If you plan on returning in the future, find "your spot" and then take a picture there each time you return - makes for a great time line (especially as kids get older).
Mark