Hi, Jennifer! I'm afraid of mascots too. Anything out of Chicago qualifies (Benny the Bull especially freaks me out.) In Disney World, I am a fan of the characters (the anomaly being Duffy. He freaks me out.)
It's easy to steer clear of characters at Disney World if you don't want that to be part of your experience. A kid doesn't usually think twice when crossing paths with the face characters (princesses, Peter Pan, etc.) It's the bigger, veiled creatures that cause pause. These characters are very good at picking up on cues to the effect of "keep your distance or my kid will flip." When my boys were your daughter's age, they liked looking at these characters from afar.
When it comes to experiencing attractions that have the potential to frighten her, I suggest starting with the more predictable, less trilling experiences. If she takes well to them, then you can gauge how well she'd do on others.
For example, to gauge how she'd do on an attraction with a boat as the vessel, try out it's a small world or Living with the Land before introducing her to Pirates. For dark rides, try Peter Pan's Flight and the Many Adventures of Winnie the Pooh before something like Haunted Mansion. I know Pirates and Haunted Mansion might sound scary at first blush, but the fear potential they both possess has never phased my two boys for a second.
That should help you get started when it comes to considering what to experience early on. Check out the
Magical Beginnings park maps for some of the most tot-friendly attractions. Your daughter will also love spending together time with you - World Showcase at Epcot and the various trails at Animal Kingdom are perfect for that.