World 1
In World 1-2, use the level's final moving platform to jump up to a new level instead of taking the pipe that normally takes you to the regular exit. Continue to the right to reach a secret exit that takes you to the World 1 Tower level.
World 1 Tower's secret exit requires you to be Shell Mario. Go through the door after you reach the level's checkpoint. Head to the upper right of the room and then further right to reach the next room. As Shell Mario, break through the bricks and go on to the Warp Cannon that takes you to World 5.
World 2
In World 2-3, when you reach the part of the level with the rising water levels, activate the switch near the coins. Swim to the right to find a secret exit that takes you to World 2-A.
In World 2-A, enter the yellow warp pipe just above the pipe at the end of the level that takes you to the regular exit. To get to that pipe, use a Spin Block to float over, using the impact from landing on top of the Koopa to send you up to the higher level. This pipe will bring you to the Warp Cannon that takes you to World 5.
In World 2-4, you need a Mini Mushroom to find the secret exit. When you get down to the water portion of the level, head left and enter the Warp Pipe to skip to World 2-6.
In the World 2 Castle, beat the boss as Mini Mario and take the bottommost path to reach the secret exit that takes you to World 4.
World 3
In World 3-2, after you find the level's second Star Coin, jump on the mushroom platform that moves just below a trio of Warp Pipes. Jump up the green one to the right to get to the secret exit to get to World 3-B.
In the World 3 Ghost House, head all the way to the right and hit the brick to reveal a switch. Hit that switch to reveal a pair of doors. The first door takes you on the path to the regular exit of the level, so backtrack to the area at the bottom of the stairs to find the door that leads to Warp Cannon and takes you to World 6.
World 4
In World 4-1, when you ride the blue dinosaur across the water, jump onto the platform made of five bricks. Stand on the third brick and jump to create a vine ladder. Climb up and enter the pipe to find a secret exit that takes you to the World 4 Tower.
In the World 4 Ghost House, after you find the level's second Star Coin, hit the switch a second time and jump on the platform it creates that leads to a Warp Pipe. Instead of entering the Warp Pipe, jump on top of it and use a Mini Mushroom to turn into Mini Mario. Wall jump to reach a platform leading to a Warp Cannon that takes you to World 7.
World 5
In World 5-2, near the end of the level, use the ice block to reach the Piranha Plant's pipe. Enter the pipe to find a secret exit that takes you to World 5-3.
In World 5-B, just before the regular exit to the level, wall jump up the vertical path and head left. Break through the bricks using Shell Mario to get to a Warp Pipe that leads to the World 5 Castle.
In the World 5 Ghost House, jump on top of the bricks at the bottom of the staircase. Jump up to hit a set of invisible bricks and then hop on top of that set of bricks. Crouch while on top of the middle brick and jump to make a vine ladder grow. Climb up the vine and wait for the Broozer to break through the wall. Go through the broken wall and then onto the floating platform. Go through the door to reach a Warp Cannon that goes to World 8.
In the World 5 Castle, beat the boss as Mini Mario. After defeating the boss, take the path at the bottom to find a secret exit that takes you to World 7.
World 7
In the World 7 Ghost House, press the Switch when you're in the room full of coins. Go left until you find a gap in the ceiling. On the left side before the gap, hit the third block from the right and climb the vine that grows out to get to the secret exit that takes you to the World 7 Tower.
In World 7-4, turn into Mini Mario and stay on the left as the level goes up. Jump on top of the flying block and jump onto the area to the left. Continue left to find a secret exit to the World 7 Castle.
In World 7-5, use Bomb-Ombs to blow up the pile of stones to reveal a pipe. Otherwise, if you're Mega Mario, do a ground pound to smash the rocks. Go down the pipe to find a secret exit to go to World 7-A.
In World 7-6, jump up on the bottom of two rows of bricks after the checkpoint. Hit the middle brick on the top row to destroy it; then hit the middle brick on the bottom row to create a vine ladder. Climb it to reach a secret exit to World 7-7.