In Chapter 1-3, pull the lamp out of the ground, but don't use it. Instead, run past the door and to the pot that's at the end of the screen. Use the lamp and go into Subspace. While inside, do down the pot. You'll be warped to Chapter 4, but you cannot switch characters until you beat Chapter 4.
In the section of Chapter 4-2 that has the whales, pull out the magic lamp that's near the beginning of that section. Carry the lamp until you find a pot and a whale that's near the bottom of the screen. Get to where they is and drop the lamp near the pot or on the whale (but you must be able to still see the pot while on the whale, or else it won't work). Go into Subspace and go down the pot. You'll be warped to Chapter 6, but you cannot switch characters until you beat that chapter.
While playing, pause the game, hold Up, B & A on controller 2 and unpause.
In Chapter 3-1, go through the door to reach the large waterfall in the cloud-filled area. Jump down and land on the platform at the base and go through the door. Go right and pull up the sixth plant from the right of the first set you reach, which should give you a lamp. Go further right and drop it near the vase to produce a warp to Chapter 5-1.
In Chapter 5-3, the area where you end the level is reached via a ladder. Before climbing down that ladder, you can squat-jump onto the overhang above the ladder. The rightmost bush of grass yields a lamp - use it near the nearby vase to create a Chapter 7 warp.