Walkthrough: After obtaining the first Power Moon, a new bridge will suddenly be created leading you forward. However, before heading across it just yet, make sure to grab those Purple Coins by diving into the waters and swimming just past the falls. From there, head across the bridge that appeared to find the Odyssey Airship. This will be the aircraft that you will ride on in hopes of finding and saving Princess Peach, the same that Cappy can power with the help of those Power Moons. In order to leave Cascade Kingdom, you will need four total Power Moons.
Continue further down the path until you reach a pole that has Purple Coins behind it. By climbing the pole, you can see a sleeping dinosaur to the left and another Broodal at the top of Fossil Falls. Head back down and head through the fossil blocks to find even more Purple Coins, straight on your way to an area to several Chain Chomps surrounding a Big Chain Chomp. By using Cappy to control the Chain Chomps, you can destroy the large fossil wall nearby. You can also crush the smaller fossil wall to the left that holds another Power Moon.
After you obtain the next Power Moon, head to the left and up the platform to find the dinosaur. Use Cappy to gain control of the dinosaur, and then use its power to break all the fossil blocks in your way and push Chain Chomps aside like they were nothing at all. Quickly break open a path to the next area, Stone Bridge, before you get ejected from the dinosaur's body.