Third Gym to Fourth Gym
Go north of town and head for the garden. You will have to fight some trainers. Fight them and catch some wild pokemon. You will eventually come to a weird/strange looking tree. Use the "Squirtle Water" on it. It will come to life. Battle it and capture it. Now go north. Do not go east or you will be going back to the first gym. You will have to fight more trainers now. Continue heading north until you get to the next town.
You will find a building where people are performing on stage. Go on to the stage and challenge all the women. They have all 5 types of Eevee. There are 2 new ones. When you beat these woman, you get a lot of experience. Go to the tall building most north and left side of the town. You will face your rival there. Also, go down the black holes. You will see some cats and items. Get all the items you can and return to the gym.
This gym uses Ghost pokemon. The platform is invisible but not hard to find out. Use this picture to help you:
Defeat him to get the phantom badge.