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. If you pre-aim the eggshot you can line up the bottom of the aiming cursor with the ground, as seen in the video to the right. Toss the egg as soon as you land on the final moving platform. 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. Aiming an egg leftward is an easy way to ensure Yoshi is facing left even if he is moving right. Backtonguing the key, as shown in the IL, is tricky but can save about a third of a second.
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.
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.
To do the 1-4 warp, begin an egg shot as you come into the room from the pipe. Hold the egg shot while you face right, wait a moment as you see in the video, and then jump off of the third log to run into the enemy. Baby Mario should go underneath the platform. Quickly you'll need to jump back across (a left right helps before the jump). Star count should go down 4 and then jump up into the pipe section and hold the jump button to do a flutter. Baby Mario will return to Yoshi, but you will be off-screen and thus transition into 1-1.