There are several things to increase your speed:

For Man:

• Jumping horizontally, instead walking
• Jump horizontally while ceilling is above you on its distance to get horizontal speedjumps, unless the height is too small or the distance for another action is too short
• Jump vertically to reach some of the platforms that are unreachable with horizontal jump
• Jump vertically then hold direction and Up to reach higher ladder
• Jump while on ladder by holding left or right to grab a platform/floor
• Jump onto enemy's opposite hitbox to avoid falling animation
• Jump onto wall to reach some higher platforms, ex. from ladder
• Rolling to avoid falling animation, unless the height is too high
• Rolling right before you get into spider-web, before the door, landing, any edge, grabbing ladder, tunnel, a cutscene or any other action, excluding horizontal jumps
• Roll to the wall and return your normal state, aka. walk again to avoid landing animation from the ladder
• Roll before an edge and jump
• Roll less closer to an edge to get faster horizontal speed
• Press both directions to avoid falling animation ex. if you failed rolling
• Hold Jump while midair to avoid falling animation, unless the height is too high
• Hold Down and press Space while grabbing a platform to fall down faster
• Hold direction, then press Up while climbing a platform to reach higher level of the ladder
• Hold both directions to avoid animations
• Hold an action while animation
• Slow down your horizontal jump by holding the opposite direction if necessary
• After grabbing a Note, hold Up and press Space to avoid menuing (can be used while rolling)
• Wait for the door/platform to open to avoid pushing animation

For Child:

• Use an action within its reach
• Use one action at a time, excluding doors and stairs
• Hold Up before a door and Down before stairs

For Knight:

• Attack enemies upwards, so you can dash optimally, excluding empowered attack
• Start attacking upwards, so that Knight will come closer to an enemy
• Attack while dashing
• If an enemy is far from a sword's length, Knight will come to that enemy and he'll still get hit
• If you dash while on opposite site to enemy, Knight will turn around
• Fall down faster when holding Space
• Use empowered attack every 3 hits. Can decrease time if you time hits correctly
• Dash while using downwards empowered attack to decrease animation time

Other strats and runs: https://www.youtube.com/playlist?list=PLuANQDhKkYZV8KEJAUiyPb435wMOjK8KY