Some rides have a pouch you can put belongings in, though they are small and cannot fit a large backpack (with the exception of rides like Dinosaur and Soarin). On FOP there is a cubby behind the bikes that you put your belongings in. On most other rides, you put your bags on the ground between your legs. On the coasters like Space Mountain, Everest and Rockin, I loop one of the backpack straps around my leg and hold it with my knees to secure it better.
There are also lockers at each park that you can rent for the day. I have never used them so I don't know how much they cost.