Walljumps and memory are the most important points here:

Walljumps:

• if you walljump, don't spam Space, but wait a little bit and then press Space once
• after you walljump you still have a second jump, which is very useful
• when you hold the opposite direction of the wall you're facing for a little bit after you walljump, your second jump will go much higher and faster
• the floating platforms have a smaller hitbox on the bottom than the sprite actually tells you, so make sure you don't fall

Memory:

• try to memorize as many rainbows as possible so you dont't have to check every corner or go back and think of efficient routes to collect all of the rainbows

Good Luck!