Warp to World 2, 3, or 4
In World 1-2, you will find some elevators near the end of the underground segment. You can use these to ride onto the blocks making up the roof, then go right and past the warp pipe you're normally supposed to take outside. This will bring you to a warp zone that can bring you to Worlds 2, 3, or 4.
Warp to World 5
In World 4-2, near the end of the underground segment, you will have to pass over a pipe with a Piranha Plant in it. Instead, you can use Big/Fire Mario/Luigi to break the blocks above it and go right and along the path to a warp zone. (Alternately, you can use an earlier elevator to get on the roof and get here.) The warp is to World 5.
Warp to World 6
In World 4-2, as you head right in the underground segment, you will see seven ? Blocks (2 over 2, with 3 to the side). Past the next gap are some blocks on the roof. You can bounce around near here to reveal some blocks, which you can use to hit the already-revealed bricks to reveal a vine. Climb this vine to go up and into a linear platforming area leading to a warp zone, which takes you to World 6.