You can make the first piston jump a couple different ways, but I like to get as close to the block I'm jumping to that I can while still being in a position to where I can just jump up & land on the soul sand the piston pushes. Once in position, you just hit the button as you jump, then sprint jump to the next platform as soon as you land on the soul sand.
For the fence & slime jump you'll want to do these in 1 run. Don't try stopping to do them individually....it doesn't work. Stand about midway on the ice block, hit the button, and sprint jump. As soon as you land on the fence jump again to the slime, and again off the slime into the web. If you made it that far, it's usually pretty easy to position yourself to fall through the doorway under the web.
The ladder jump isn't all that hard, but you can easily fall off the ladder after you hit it if you don't quickly shift & turn back to the left.
The first water jump is easiest if you use a double jump & take off from the very edge. Jump as far as you can, and you shouldn't fall through the water. The second is easier, but again, jump as far as you can hitting the water as high up as you can. The last one is the tricky one. It's just a matter of lining yourself up with the doorway, hitting the water & getting far enough. Sometimes it seems like you make a perfect jump, but you still fall. But normally you just weren't lined up perfectly when your feet passed the bottom of the door.