There are free-to-use lockers (for two hours, I believe) at the entrance to Kali River Rapids in Animal Kingdom. Almost all other rides on WDW property are designed for you to bring your bags and personal items on the ride with you. For the more "exciting" rides, you stow your bag at your feet and they'll often include a pouch in front of you to stow sunglasses and hats, etc.
FYI, when getting on a roller coaster, I *always* ensure my phone is either in a zippered pocket on my person, or inside my bag.