Continue on the same world you got a Game Over on:
If you get a Game Over, Hold Select, and when it gets back to the title screen, press start to begin you game.
You will continue on the World you got Game Over on.
(Not the level, unless you died on the first level of that world.)

Fireworks:
At the end of a level, touch the flagpole when the last digit of the timer is a 1,3, or 6. You will get a fireworks in the amount of the digit. You also get points per firework.

Minus World:
The famous Minus World. To access it, go to World 1-2. At the end where you can go into that tall pipe to get out of the underground, brake all the bricks you can above it except the last one touching the pipe.
Then stand on the pipe, and face opposite of it. Then jump backward, hopefully you'll go through the wall, and be in an area with a pipe leading to the Minus World. Don't worry if you don't do this right away, it's a  little tricky.

The Warp Zones:
First Warp Zone, Worlds 2,3,4: This Warp Zone is at the end of World 1-2. Near the very end, with the last platforms that go up, jump on them and get as high as you can, then run and jump off. You should land above that big pipe at the end. Keep going straight and you'll be at the Warp Zone.

Second Warp Zone, World 5: This Warp Zone is at World 4-2. At the very end of the level, get to the top of the level, above the bricks. You can do that earlier in the level by using the platforms that move upward. Or standing on the tall pipe at the end, and braking some bricks above you, then jumping up.

Third Warp Zone, Worlds 6,7,8: This Warp Zone is also in World 4-2. Near the very start, after the first set of moving platforms, you'll see a line of 3 bricks randomly in the air. There is a beanstalk in one of them.
There are 3 invisible blocks you can hit and reveal below them to get up to the beanstalk.
Hit the invisible blocks and get the beanstalk, take it up, and be in an area with coins and then 3 pipes to worlds 6, 7, and 8.