To be able to go to World 9, you must beat the main game without the use of any warp pipes.
Access Worlds A-D
To access Worlds A-D, beat the main game (Worlds 1-8) eight times, then press B+Start at the title screen.
Continue
If you hold A and Start after getting a Game Over, you can restart at the world in which you died, so long as it is not Worlds 9, A, B, C, or D.
See the Famicom disc system intro
When you turn on your Game Boy Advance, hold A+B at the same time to see an intro from the Famicom Mini version, which is Mario and Luigi fighting over a lightbulb, and if you press B again you will get to the loading screen from the Famicom Disc system, all of which were just added as an easter egg for the GBA version that you can not normally see on this game.
Warp to World 1 (from later levels)
In World 3-1, at the end of the stage use the springboard to launch past the flagpole. The warp zone following takes you to World 1.
Warp to World 2
In World 1-2, get on the bricks at the top of the level as soon as possible. Stay up there and eventually cross past the exit pipe. Eventually, you'll just be able to keep running right to the World 2 warp zone.
Warp to World 3
In World 1-2, pay attention to the surroundings. When you see a pipe, several bricks, and a Buzzy Beetle nearby, hit the first of these bricks to the left. A vine will pop out: climb it up and this area will take you to the World 3 warp zone.
Warp to World 4
In World 1-2, go along until you see some moving platforms. Get on them and use a running jump to get to the small brick ceiling. Jump across the gap to the right and keep going until you see two pipes. Use the left pipe to find an area with lava. Get small with your character (in other words, get hurt if you've picked up a power-up) and go across the small lava pit under the brick wall. Keep going past there to a pipe. Ignore said pipe, but jump around until you find the hidden coin block nearby. Jump onto it and step carefully to its right edge and jump: you should find another block. Jump on this and up to the ceiling, then continue along to the warp zone.
Warp to World 5 (from later levels)
In World 8-1, after passing three Hammer Bros. go down the next pipe you can. In this water-filled area, go along the path to the exit. Outside, go upstairs to the warp zone.
Warp to World 6
In World 5-1, when you're Super Mario/Luigi, you'll see two rows of floating bricks past two Paratroopas near the end. Hit the rightmost brick on the top row and climb the vine that is released. After the small bonus area, you'll be dropped off on a pipe next to a warp zone.
Warp to World 7
In World 5-2, at the three tiny moving platforms, use said platforms to reach the ceiling. From there, run to the right, jumping wherever the ceiling ends, and you'll eventually reach the warp zone.
Warp to World 8
In World 5-2, continue until you reach the moving platforms that number three. There, hop on the first and use it to hit the brick to the left of the platform's path: a vine will pop out. Climb it to find the warp zone after a small bonus area.
Warp to World B
In World A-2, about halfway through the underground part of the level, you'll find a set of staircase-like blocks. You can break through the blocks above them and get on the upper portion of the area. From there, you can run and jump heading right and past the warp pipe heading outside - this takes you to a World B warp zone.
Warp to World C
In World A-3, near the end jump high on the springboard and fly rightward. Once past the flagpole, it's easy to reach the warp zone to World C.
Warp to World D
In World B-4, take the very last pipe in the level, the one next to the lava. This takes you easily to a warp zone.