Like the original, platforms appear out of the blue lava so you can cross, but the difference is that the platforms are sparse, so you will need to be jumping a lot and might need to utilize the Tanooki Suit a lot more than usual. At the beginning, jump across the first set of blocks that appear and hit the ?-Block for a Super Leaf, making sure to watch out for the Splorch. When you get to the next set of blocks, one of them will have Green Star #1 on it. Collect it, then continue until you reach the platform with some Parabones. There will be another ?-Block with a power-up here, then you will have to cautiously cross the rotating spike rollers.
Note that you can actually stand on top of the spiked rollers platform without getting hit, which is much more stable than trying to navigate the appearing blocks. In fact, you will need to jump on top of the second spiked roller to grab Green Star #2. Continue to the stable portion of the level, where you will find the Checkpoint Flag. You will be going on left to right now, across the blocks coming out of the walls and on circular platforms with spiked rollers rotating around them. At one point, you can grab another power-up from a ?-Block, but you will also have to contend with some Lava Bubbles.
Once you reach the end, do not go in the green pipe just yet. Instead, if you take a glance to the right of the pipe, you should see a cloud platform with Green Star #3 on it. Use the Tanooki Suit to float over and collect it, then return to go down the pipe. The last area will just consist of more blocks coming out of the lava, then the Goal Pole.