The biggest trick in this level is causing a sprite overload by creating a bunch of stars with a big egg. This can despawn the bullet bill launchers, allowing you to skip ground pounding the first set of posts. To perform this trick, fire a large egg as shown in the picture. Make sure you have 4+ eggs (of any size) on Yoshi, as more eggs makes the despawn easier.
Jumping onto this platform (as opposed to fluttering over it) seems to make the despawn easier (possibly by ensuring all of the stars are on-screen when the bullet launchers are scrolling on screen).
The whole process illustrated:
It is possible to flutter under the 2nd falling rock in room 2 with the right camera manipulation, but it is much safer to wait and jump over it.
To get over the midring in the final room, wait for Yoshi to come to a complete stop, then press jump and right at the same time. If you go too fast, you will collect the car bubble when you walk under it since it will be at a low point in its bouncing cycle.