At the Magic Kingdom It's a Small World, (if it has reopened) Peter Pan's Flight, Cinderella's Golden Carrousel, Dunbo, Tea Cups (just don't spin to fast) Many Adventures of Winnie the Pooh, Counrty Bear Jamboree, Goofys' Country Dancin Jamboree, Tom Sawyer Island, Liberty Square River Boat, Walt Disney Railroad, Swiss family Treehouse, Magic carpets of Aladdin, Jungle Cruise, Indy Speedway, Tomorrowland Transit Authority, Carousel of Progress,and Buzz Lightyear Space Ranger Spin.
At MGM Voyage of the Little Mermaid, Playhouse Disney, Beauty and the Beast, Jim Henson's Muppets 3-D, Honey I Shrunk the Kids (playground) (most of these are shows but they have music and most baby's love music)
Some are also dark like Peter Pan Flight so if there is a problem you may want to skip that one, my nieces and nephews loved all the above attractions when they where as young as your child but every child is different don't take him/her on anything dark first start with Dumbo, Pooh and the Tea Cups. Then see how he/she does the Indy Cars where a favorite of my little ones because they try and drive them with you being there for the gas and braking.
There are many things for the little ones to do and have a great time...also be sure to ask about the baby swap options for each ride then everyone can enjoy the rides.
Riding on the Monorail and buses where another big treat.
Remember to a 12 month old...Mickey is a Seven Foot Rat. Let your child decide about the characters.