The fastest beginning route is to go along the bottom, though the top is a bit safer. If you choose the bottom path, be prepared for a randomly spawning flame at the very end. A neutral egg as you're jumping into the final platform before the save ring should be fired just in case.
A neat trick featured in this level is the off-screen pot break to get the key quicker. If you're able to hit the pot with an egg while it is still offscreen, it will instantly shatter, removing the need to push the pot off its ledge. The key will then fly in the direction Yoshi is facing, so you want to face away from the pot to have the key fly towards you.
It's also recommended to jump off of the mini-Burt, skipping the middle ring, just before the boss door. A single tongue (Y-press) as you pass over the middle of the blue door gives you just about the right amount of slowdown to seamlessly jump off the enemy, rather than using left+right as shown in the video at the end of the section.
Landing the first egg shot on Burt as quickly as possible can be difficult (if your egg control is set to hasty, spam A at the start of the fight, but stop spamming once you see Yoshi start the egg aiming animation).
Burt can occasionally be knocked directly at you, depending on where the egg is in his body when his invulnerability ends. Be prepared to move away from him, or give yourself enough room in case he does get knocked towards you.
This level has many difficult egg shots and tricks. To start, collect as many eggs as needed from the beginning egg blocks. A backwards tongue is the fastest way to grab them.
The first flower can be difficult to hit between the burts. The trick is to push left briefly when bouncing on the arrow, to force the screen to scroll slower. This makes the burts spawn later and gives them a more favorable position for shooting the flower. You can use a visual queue like the notches in the wall or evenly spaced jumps to help time the egg shot.
Consult the any% section for the offscreen pot break.