Continue
When your game is over, hold A and press Start on the title screen. This lets you continue from the beginning of the world that you were defeated in.

Bonus Fireworkds
If you touch the flagpole when the last digit of the timer is a 1, 3, or 6, fireworks will go off, giving you 500 points each. The number of fireworks you get matches the last digit on the timer.

Multiple 1-Ups: World 3-1
In World 3-1, at the staircase before the flagpole, there are two Koopa Troopas. Let the first one by, and get all the way to the right on a step as you can. Jump straight up to stomp the second one. If done correctly (a quick, short jump usually does the trick), Mario will hit off the Koopa Troopa, and will start a cycle of bouncing the shell off the step and landing on the shell again. If you don't touch the ground, your points will rack up, and you'll eventually get 1-Ups. If for some reason you land again, just press A and the cycle will start again. Try not to get more than 128 extra lives in reserve or your game will end next time you lose a life.

Warp Zone 1
In World 1-2, go over the end pipe by walking on the ceiling, and behind the wall you'll find a Warp Zone where you can warp to World 2, 3, or 4.

Warp Zone 2
In World 4-2, walk on the ceiling like you did in 1-2 and you'll find a Warp to World 5.

Warp Zone 3
This is the second Warp Zone you can reach from World 4-2. After the first elevator lift, there is a pit and three brick blocks above you. Jump around, and you'll find four hidden coin blocks under the three bricks. Use the hidden blocks you just uncovered to hit the first brick which will sprout a Vine. The Vine leads to some Coins, and the second Warp Zone.

Minus World
World -1 (A.K.A. the "Minus World") is an infinitely-long water level, only accessible through World 1-2. Here's what you do: As Super Mario, get all the way to the green pipe at the end of 1-2. Stand on top of the pipe and smash the blocks above you. Do not smash the very last block on the right. Now, stand on the end of the pipe and face left. Jump straight up, then press and hold Right on the controller. What you are trying to do here is jump through the bottom left corner of the block backwards. If you do it just right, Mario will slide through the block and be pulled through the wall. When Mario gets to the other side of the wall, go down the first pipe before it says "WELCOME TO WARP ZONE." You will not come out in 4-1, but in -1. Yes, that's right, -1. World -1 is just like 2-2, but the pipe at the end takes you back to the beginning.
Side note: If you go down them before it says "WELCOME TO WARP ZONE," the second pipe takes you to 5-1, and the third pipe also takes you to the Minus World.
Final Note: Rumours of a second minus world have been greatly exaggerated.