At the start, if you need the 4 eggs, go with the route in the video. If not, an alternate route that runs to the right first before getting on the balloons is marginally faster, as it does not require fluttering (which is slower than running). Jumping off of the balloons continually helps them raise quicker.
Bouncing on the bullet bill in the next section helps you travel vertically much quick. The key to achieving maximum vertical distance is to hold down jump prior to landing on the bullet bill.
The section just after the bullet bill is quite tricky to learn. Make sure you are releasing the dpad if you choose to spit koopa shells at the green pirahnas, as you will lose speed otherwise. If you have an egg, and easy method of doing the final green piranha is to either a) jump and tongue the piranha's projectile, then fire off a quick egg shot to pass through the piranha while holding right the entire time, b) jump under the projectile and then do the same.
It's possible to reduce some lag at the section right before the goal ring by spitting a shy guy or eating one of the 1up baloons.