Hiya Jen!
The safest place is on your person. Unless there is someone in your party waiting outside, then you can dump it all on them, and run away before they can protest!
I like to wear one of those "cross body bags" while walking around the parks for park essentials and my camera. It generally is ok on me when I get on a ride. Sometimes, on a ride like Soarin' where it has a storage bin underneath the seat I stow it there because I feel it's safe....but that is rare!
The more you sit a bag down, the more likely you are to leave it! Disney has a great lost & found, but it's better to avoid it if possible!
If you are bringing a camera with you, stash a plastic ziplock baggy in your bag in case of rain or a water ride so you can protect it.
Have a wonderful time in Disney World!