With the warp zones in Super Mario Bros. you can skip entire worlds. In World 1-2 there are three tubes to choose from. World 4-2 has two separate shortcuts: the vine path to World 6, 7, or 8 and a single warp tube to World 5.
For all 32 levels without world warp, open the complete walkthrough. The following paths apply to the original NES version. Super Mario Bros.
World 1-2: Warp zone to World 2, 3 and 4
Play to the two platform lifts at the end of the cave. Platforms move downwards on the left and upwards on the right. Get on the right lift, let it carry you up and jump onto the ceiling. Continue to the right above the normal exit pipe.
The warp tubes are in the room behind. From left to right they lead to World 4, World 3 and World 2. The number above each tube indicates the destination. For the quick route to Bowser, take the left pipe to World 4.
World 4-2: Vine path to Worlds 6, 7 and 8
After the first downward elevator you reach an upper row of bricks. There are invisible stepping blocks underneath. Hit it from below so you can reach the left brick with the Vine. As big Mario, be careful not to break the desired jump path.
Climb up the vine and follow the upper bonus path. The three warp tubes at the end take you to World 6, World 7 or World 8. For the direct route to the final world, choose the tube with number 8.
World 4-2: Second Warp Zone to World 5
If you skip the Vine, you can continue playing through the tunnel to the normal exit tube. Go above this pipe instead of entering it. Behind it lies a single warp tube to World 5. This exit does not lead to the warp zone in World 8.
The short route to Bowser
Play 1-1 → 1-2, use the warp to World 4 there and play 4-1 → 4-2. You get to World 8 via the vine route. After that you have to do 8-1, 8-2, 8-3 and 8-4 regularly. For the last tube path, the solution to Bowser’s Castle in World 8-4 helps.