Magic Kingdom- Dumbos, Pooh, Small World, Peter Pan, carousel, TTA, Buzz, Jungle Cruise, Aladdin, train, characters are great with 1-yr-olds.
Epcot- Living Seas, Living w/ the Land, Mexico ride, Spaceship Earth (if your back can handle the extra weight of holding the baby-ugh.) DD loved the fountains and watching the coots on one of the ponds.
MGM- Maybe Playhouse Disney and Little Mermaid depending on how your child sits through shows. I personally didn't think MGM had much to offer the little ones other than tons of characters. Oh, and if mobile enough, the Honey, I Shrunk playground is great.
AK- Triceratop Spin, safari, animal trails, bird show is great if your child can sit through shows. We didn't take our show-hating kids to Lion King or Tarzan but my parents said they were pretty loud, though THEY enjoyed them. AK also has many characters.
My brother just got back with his DD just turned 1 and she loved all the gentle rides and hated all the shows/3D movies that were loud. She really loved the characters and many of them made a huge deal over her. She doesn't even stand yet so they would get down on the ground and play with her. When we took my DD almost 1 1/2, she did great with everything except eating. She almost exclusively at ketchup (with a spoon.) Yuck!