Mario Bros Guide:

Don't worry about using warp zones they won't harm your game.

Warp Zones: The 1st Warp Zone is in World 1-2 At the end of the level where the pipe leading back to the serfice.
don't go through the pipe instead jump on the platform behind the pipe again and leap on the roof of the cave there is a small space
which you can fit through. once up on that roof just keep runing.
(You will run over the pipe.) You will now see three pipes with pirana plants in them but don't worry they will disapear when you jump on there pipe.

The 2nd Warp zone is a bit harder to reach it is in World 4-2.
Near the start of the level just after where the moving platform is there is some invisible blocks uncover two these two:

There are three blocks above the two you just uncovered.
Smash the middle and the right block then hit the left block to reveal a beanstalk:

Now simply climb the beanstalk to reach a new floor. jump on the mushroom platforms for some coins then run for a few seconds to reach the warp zone:

The thrid warp zone is in the same level as the second warp zone, World 4-2. Like the first warp zone you have to get on top of the pipe leading to the surface. this time there are three places you can get into the alcove on the roof.

Here is the first place:

Here is the second place:

And here is the third place:

You will need to smash that gap in the ceiling over there.

You know the drill, just run over that pipe into the warp zone.

Here is a map of the final level:

It may look not that hard here but it's full of traps that you can only just see on this map!