Started doing some things on Four Island. I noticed that there was a Day Care located on the island, and heard some things about eggs being found there. Inside a house, I met this bragging jerk guy who gave me a sticker for my Trainer Card for entering the Hall of Fame. Other than that, by talking to many of the other residents in town, I learned that Lorelei of the Elite Four is from this island.
When I was done in the main town, I Surfed over to the entrance to Icefall Cave and headed inside. I saw a giant waterfall shortly after entering the cave, though I had not way to climb up it so I went deeper into the cave instead. I had to deal with some ice puzzle stuff and eventually found the HM for Waterfall. I returned to the first room of the cave and taught Waterfall to an HM slave I brought with me.
I climbed up the larger waterfall and then entered another room. There, I saw Lorelei and several Team Rocket members who were causing problems for the Pokemon in the cave. I helped Lorelei out by fighting and destroying one of the Rockets in battle, then they all fled for their hideout on Five Island. After that, I made my way back to the town.