World 1-2 Warps
In World 1-2, as you pass over the rising and falling construction girders at the end of the level, jump onto the one that's rising to the top of the stage running above the warp pipe that normally leads to the flagpole. Keep running across this until you come to a drop zone with three warp pipes. These pipes will take you to Worlds 2, 3 and 4. As you traverse the girders, don't let the rising girder carry you all the way to the top of the screen because it disappears.
World 4-2 Warps
The warps from World 4-2 require you to climb back into the overworld. After you pass the second mushroom where the girder moves up and down, jump up toward the rightmost block. This will reveal a hidden block. Another one can be found to the left and above, and a third one to the left and above the second block. You should be just below the leftmost visible block. A vine comes out of it leading to the overworld. Make your way to the alcove of orange pipes. These take you to Worlds 6, 7 and 8.
1-Up Trick
The higher levels of the game greatly increase in difficulty. You can rack up extra lives beforehand in World 3-1 at the ending staircase. As you approach, you'll see two Koopas coming. Jump over the first one. Jump on the second one so it bounces against the wall, and continue jumping on it to keep yourself airborne. As you do this, your points add up until you start getting 1-Ups. Don't go over 99 lives with this trick--otherwise, it's game over.
World 8-4 Solution
Getting through World 8-4--Bowser's Castle--proves difficult to many first-time players because it loops unless you know the pathway. According to GameFAQs, you can travel through all the pipes, but some of them take you back to the beginning of the level. In the first section, take the third pipe. In the second section, take the pipe floating on the midair block. Continue through the third section, avoiding the second pipe. Take the one past the large pit. Traverse the underwater section to get to Bowser.