In a single-segment run, entering this level with at least one egg is important to save time hitting the balloon carrying the bouncy arrow faster. Most of the rest of the level is straightforward, with small optimizations here and there like jumping as soon as possible while riding the slow yellow lift.
Near the end of the level, an extended flutter can be used to skip the beanstalk, saving a few seconds. There are many ways to get the enemy to land on the mushroom platform but an upspit as shown is likely the simplest:
Acquiring as many eggs as possible now helps later in 1-8.
1-6 is a level where egg order starts to become increasingly important. It also is the first level to introduce the Flashing Red Eggs. You can simply complete the level with a flashing red egg still in Yoshi's stash, or throw it and pick up the red coin that is dropped. This can be risky, however, as missing the egg shot getting all 20 red coins impossible.
There is relatively accessibly star cloud at the very end of the level, however it is faster to forgo it and use red eggs to obtain the remaining stars needed. This requires creating a red egg at an area such as the slow moving platform to access the outdoors area, and switching between eggs so that the red eggs are used on something like the red switch cloud, as demonstrated in the movie/gif.
The fastest way to skip the beanstalk at the end of the level is to use this method: the screen must stay scroll to the left for the bat to spawn.
A few different ways to get some stars if needed: