You can take your backpack with you on pretty much any ride. Many rides have a place to stow your bag. For those that don't, you can set your bag at your feet. If you go on Rock n' Roller Coaster, just set your bag at your feet and loop one strap around your ankle (RnR goes upside down, so you'll want to be sure to loop that strap around your ankle).
The only ride where you'll want to stow your backpack before going on the ride is Kali River Rapids. The ride cars have a small place to stow small items in the center of the car (to keep things dry), but there's only one fairly small storage bin for several people to share. At KRR, they do have storage lockers available just outside the ride that are totally free for two hours. So if you have something as big as a backpack, store it in the KRR lockers.