The first bit is easy, just don't jump from a half slab when you're clearing the gap, and shift when you get close to landing just to be safe.
The iron bars are easy until the last jump. Just remember you CAN fall through the middle of the squares. Use shift as soon as you land to make sure you don't slip through the middle. The last jump is a head bumper, which is the only challenging part.
When you get to the slimes sprint jump onto the first & don't stop until you get to the 2nd. When you're about to land hit shift & hold back. Your momentum & the slimes natural bounce can send you right over if you're not careful. Make an easy jump to the 3rd slime, no sprinting required. Then get on the right of that block so you don't have to turn as much in the air for your next jump. Don't just fall onto the 4th slime, jump & use it's bounce to take you to the 5th slime.
For the piston boost, just shift & stick your butt toward the platform you want to land on. Hit the button & as soon as you're airborne, hit the back button to land on the platform.
Don't hit the button in the wall...trust me. And for the last jump just do a simple double jump.