1-2 Any%

Basics

The major notable trick in this level is referred to as the "1-2 skip," which entirely bypasses the second screen of 1-2
Though very precise, the trick is easily attempted multiple times in succession and can be grasped by a new player with a little bit of practice.
The recommended method for performing this trick involves spitting a flower enemy (Crayzee Dayzee) into the wall and bouncing off of it as shown in the GIF.
After landing 1-2 skip, you go a bit to the left so you can have right momentum when you clear the wall.
You have to jump over the second pit or you will fall in and lose time.
Be sure to have an extra egg to shoot the pihranha plant near the end, saving a bit of time not collecting the red coins.

Spit Method

You set up the flower spit by touching the left side of the wooden post closest to the wall as shown in the video.
Then jump onto the post while holding right and jump when you are in about the middle of the post.
Hold B the entire time.
When Yoshi is about directly over the red arrow, release forward and press Y at about the same time.
If you did it on the correct frame, you will continue holding B and be able to easily flutter over the ledge.

Perfect Jump Method

This 1-2 skip method is performed by combining a perfect jump with an extended flutter.
The trick is simple to understand, but hard to execute: you must bounce on a Shy Guy and then immediately perform a first-frame jump off of the wooden posts.
If executed correctly, Yoshi will flutter much higher than usual.
A second flutter can then give Yoshi just enough height to go on top of the level, and skip the entire middle section.

You can opt out of 1-2 skip and complete the 2nd screen when just starting out.
For the jump under the flower, just press and hold jump at the red arrow sign and the auto flutter will get you on the ledge.