2-6 is a very straightforward level that doesn't have any particular tricks associated with it so just refer to the video for the route. It's highly recommended, however, to end the level with at least two eggs (ideally three or more) because you'll be using them to save a little bit of time in 2-7.
In the first area, be sure not to run out of eggs before spawning the door, or you can get trapped and have to reset.
In the cave, proceed to the right and hit the bouncy arrow. To gather the top red coins there, simply fire an upward shot from the third wooden platform from the bottom (you can also get them with an upshot from the first platform if you hold up until the screen scrolls).
For the foam containing the door cloud just after the goal ring, if you start an upshot and jump into the lower part of the goal ring then release the egg shot as your stars are going up, it is possible to hit the cloud in the foam, as demonstrated in the gif.
In the bonus room, the red coins/flower only spawn after the shyguys have been killed. If you jump on them, they do not count as being killed until their corpse fully falls off the screen. Thus it is faster to spit them into each other because they explode.
The section directly after this can be done as shown below:
The level is pretty straightforward after that. After collecting the final flower, it is quicker to jump up and over that area rather than going down underneath.