This comes first regardless of route because of the boot capsule in the stage. It's also one of the easier stages in general, with some optional tricky strategies.
First time through[]
Before dash upgrade[]
Until you get the dash upgrade, there's little variation in strategies to do, there are some optional slope jumps that save a small amount of time. In general, just run to the right and shoot everything before you run into stuff. The rabbits die to 2 uncharged shots, or either of the charged shots. The lumberjacks you should be killing by using a full charge shot then mashing shoot to kill them.
After dash upgrade[]
After you get the dash upgrade, dashing will be the primary method of movement for the rest of the game, and options for strategies opens up a lot.
No stop hallway
In the section immediately after the dash boots, there are a couple options for how to get past the first flamingo with the two flyers around it, the fastest of which is usually referred to as the 'No stop hallway' which uses a full charge shot and 2 dash shots to get a double kill on the flamingo and one of the flyers to get through smoothly. That method has a 2 frame window to get the doublekill, though, and requires some fairly consistent movement and one precise shot, so using other methods might be easier to get into runs. It doesn't cost a lot of time if messed up, and the stage is relatively easy, so as far as difficult strategies go this one is a good one to pick up if you don't mind taking a hit once in a while until you really get used to it.
An easier way through that section is with a dash wallkick off of the wall below the ledge the flamingo is standing on and using several uncharged shots to mow everything down, if you get the wallkick low enough, the lower of the flyers will actually be so low it misses you entirely as you jump over, so you don't even have to kill all 3 of the enemies (this method is the one used in the video at the top of the page).
Beyond the first flamingo, just jump over the rest, theres only one flyer that can get in your way, but you can kill it with just about anything, dash shot, mid-charge shot, charged shot.
Snowball section damage boost
The next difficult section is the snowball section, there are a lot of projectiles and enemies to avoid, so it can be tough to do quickly. The most simple way to get through uses a damage boost from the second snowball thrower (getting hit to the right, hopefully) to progress through some of the other enemies on your invincibility frames, and using charged shots and dash shots to kill enemies that you can't avoid killing.
Snowball section
"Zam's balls" strategy, thanks philthyturtle
The fastest way through is avoiding taking the damage boost and instead using a mid-charge shot plus a dash shot as you jump over the second snowball thrower, then dash jumping as soon as you land and using 2 more dash shots to kill both the snowball and thrower in your way. From there, use well placed dash shots to destroy snowballs and throwers until you get to the door.
Chill Penguin fight[]
Dash shots deal 2 damage, mid-charge shots deal 2 damage, and full charge shots deal 3 damage. Open the fight with releasing a charged shot. You want to hit penguin with either dash shots or charged shots, since mid-charge shots deal the same damage as dash shots, and having a mid-charge shot on screen prevents you from charging. When penguin breathes out ice statues, you have an opportunity to use a Cf-0 strategy if you want to, where you get as far away from penguin as possible, shoot a dash shot so it hits him right as he becomes vulnerable, and hold a charge until you can hit him with a fully charged shot.
The bad behaviors penguin can give are sliding and spitting ice chunks. Sliding is by far the worst, since he's entirely invincible for the duration of the slide. Use his sliding time to charge up a shot and try to hit him as he comes out of it. Spitting ice chunks isn't nearly as bad, but the ice chunks can and will block any shot you fire into it. If you aim a shot toward his head rather than his body, even if it isn't timed exactly right, it can avoid the ice chunk and hit penguin anyway, but it's also very easy to shoot too high and just completely miss.
100% revisit[]
Penguin revisit, basic
Basic
Penguin revisit, no tornado
Faster, no tornado revisit
You need to revisit specifically for the heart, so technically this can happen any time after you complete mammoth, but it's best to wait until at least after Chameleon so you have c.sting to make things easier. The traditional way to open is with tornadoes on the lumberjacks. This is a bit slow, though, since you have to wait in front of the lumberjacks while firing the tornadoes. Other openers include using fire, spark, or just the buster through the first section. Either way, swap to c.sting for the cave, kill some of the bats on the way down and charge up a c.sting shot and use it to blow past all the wheel enemies easily.
Use another charged c.sting to get past the flamingos, then swap to fire and get the heart.