Grab as many of your crab children as possible as you ascend the tower!
Don't be too quick to head to the top, and if you just miss out on grabbing a baby crab, keep traveling - more often than not there'll be a wall coming up that you can bounce off to change direction.
Pay close attention to what's coming up around the next rotating corner.
Make sure you bop off the heads of any enemies, otherwise you'll be sent dropping to a lower level - of course, you can always use that as a tactic to descend quickly if you've missed out on any crabs.
You never have to do anything other than bounce off the heads of the gigantic bosses in the game. Keep repeating that and they'll eventually expire and drop their precious hat.
Often you'll have to get to a decent height in order to drop onto them - head straight at them and you'll likely be knocked downwards in the stage.
Don't forget that when you're knocked down to a lower level by an enemy, you'll also lose your crab babies.
Very often the level will rearrange itself to open up new pathways, but you'll need to trigger the change by performing a certain action. That might be traveling over a certain part of the landscape, or collecting a particular bubble or baby crab.
The arrows aren't just useful for pointing you in the right direction, they're also excellent markers for where you should begin your jump. If you're struggling to clear multiple gaps that appear quickly one after another, follow the arrow guideline for your starting jump and you should clear the obstacle cleanly.