Hi Elizabeth! Congratulations on your baby's first birthday! These times are so special, and there's no place I'd rather have a birthday than at Walt Disney World! You would be surprised at the selection of attractions that you can take your one year old on. Most of the rides at Magic Kingdom are baby friendly, save of course for Space Mountain, Big Thunder Mountain Railroad and Splash Mountain. At Epcot, she will be able to enjoy Spaceship Earth, Living with the Land, Journey into Imagination with Figment, Three Caballeros and all the awesome shows that are offered at World Showcase. At Hollywood Studios, she can enjoy Disney Junior--Live on Stage, Voyage of the Little Mermaid, Muppets 3D and Beauty and the Beast Live on Stage! In Animal Kingdom, she will love Finding Nemo--The Musical, It's Tough to be a Bug and of course Festival of the Lion KIng! These are only a handful of the attractions in which she can ride with you.
For her special 1st birthday, you can definitely get her a first visit and first birthday pin at City Hall in Magic Kingdom or you can plan a special dinner with Cinderella at Cinderella's Royal Table. The options are endless for what you might want to plan for your little girls special day!
One of our wonderful Moms, Amanda A., answered a questions similar to yours about First Birthdays at WDW. She gave such an awesome answer that I am linking it for you below.
First Birthday at WDW
Hope this helps you plan a wonderful first birthday and vacation for your daughter and you!
-Malanie