Cheat: Warp Zone Pipes
You will find the first Warp Zone located at the end of level 1-2. In order to reach it, you will need to ride the previous elevator to the top and then jump onto the brick ceiling. Run all the way to your right and you will find the Warp Zone. Enter the corresponding Pipe to skip to world's 2, 3 or 4.
There are more Warp Pipes scattered throughout the game, but it's 2013, you should know where these things are.