I made some progress today. First I went back to the Deep Jungle and cleared it completely. First I had to find some slides for Jane so she could show them to Tarzan, so he can, in turn, possibly say he saw one of the things on the slides in relation to Kairi or Rikku. It didn’t help, but Tarzan took us to his Gorilla pack and asked them to reveal their nest to us. They refused so we continued back to the three house. There, we’ve found Clayton trying to shoot one of the Gorillas so we stopped him. As we came back to the tent, Clayton got pissed and left, and the Heartless started attacking.
After clearing most of the areas of the Heartless, I came back to the cliffs where I Clayton attacked us, being controlled by the Heartless. It was a pretty tough boss battle and I had to use a few potions, but we managed to beat it. As a reward, I got the Cure spell and Tarzan took us to his home behind the waterfalls. We found another Keyhole there and unlocked it, which granted us another Gummi part. As I was leaving the Deep Jungle, Tarzan gave me a keychain which increased the reach and damage of my Keyblade.
First I went back to Traverse Town and explored it for a bit, but then I decided to go try and kill Cerberus again. With the cure spell and the new Keychain, as well as a few more levels, I killed him easily in one try. I guess this is how the game is supposed to be played. Most of the times you have several options on where to go next and nothing is stopping you to explore another option if you get stuck somewhere. Anyway, this is where I stopped playing and decided to continue tomorrow, so I guess that is what I will be doing.