My husband and I went for our honeymoon in mid-June. It wasn't too bad. Just slightly hot, with an hour's rain about 4p, and the crowds weren't terrible. We avoided the roller coasters except for Splash Mountain because he can't ride them.
I recommend all of the parks. Each park has a few rides, a lot of shows, and a lot of cool things to look at. MGM is attached to DisneyWorld, and they have movie-based shows, like Indiana Jones, Muppets, and a Star Wars ride. And if you like zoos (who doesn't?) hit the Animal Kingdom. Not only do they have animals in semi-natural habitat (think Asheboro Zoo), but the Tree of Life is detailed and can keep you occupied for at least a half hour trying to pick out animals. Epcot was pretty cool, but the best best best part about it was the World Showcase. We had to go there twice so we could eat more food.
Missing the afternoon parades is a good idea if you want to hit the rides, but I recommend hitting at least one of them. If you spread the parks around, try hitting the night shows. Most of the rides are closed or closing by then, anyway.
Downtown Disney.. avoid if you're trying to save money.
Though there are some really cool stores there. The Lego store has some life-sized lego constructions outside which is REALLY cool. DisneyQuest is one big arcade with state-of-the-art games. It was also the only place we could find to get online and say Hi to everyone and send emails.
I can't think of anything to particularly avoid. We spent the whole week there, pretty much on the go every day, and we never got bored. We'd like to go back someday. However, my dad took the family there a few years ago, and my brothers (12 and 7 at time time? somewhere around there) didn't like it and were bored the whole time.
