Warp to World 4
In any of the first three levels of the game, go to the area right before the brick background on the screen. Keep picking up the plants until you find a bottle. Once you find the bottle, go past the brick background until you find another pot. Throw the bottle on the pot, and you will warp to World 4.
Warp to World 5
Beat the game until you reach level 3-1. Once you are there, enter the first door and fall all the way down. When you land, you will be on a platform with a door. Enter the door and pick up the potion hidden in the grass. Throw the potion to the ground and enter the door that appears. Through the door, walk into the vase, and you will warp to World 5.
Warp to World 6
Beat the game until you reach level 4-2. Enter the area with the whales, find a bottle and throw it at the pot in the middle of the screen. Enter the doorway and walk into the pot to warp to World 6.