Basic Movement
While Hat Kid's moveset is relatively small compared to other platforming heroes, it is a highly versatile one that allows her to chain together many moves and reach most areas with relative ease and style!
Crawling
If the crawl button is pressed while she's stationary, she gets into a crawling position and can enter through small tunnels.
Jumping
if the jump button is pressed, Hat Kid will jump into the air. Pressing it again while airborne will perform a second jump. While this is a basic ability in most platformers, jumping and double jumping is the foundation to Hat Kid's impressive platforming skills.
Dive
If the crouch button is pressed while she is moving, Hat Kid will dive in the direction she's facing. This briefly increases her speed and can be performed both on the ground and in the air. While diving, Hat Kid can change her direction very slowly. Dives can be canceled simply by pressing Jump or Crouch, which reverts her back to her walking speed.
Diving in the air will briefly slow Hat Kid's vertical speed and lunge her in the direction she faces, giving her more movement horizontally. Jumping to cancel a midair dive also briefly slows her vertical speed. This allows the player to control Hat Kid in the air on a wider horizontal plane and allows her to land more precisely on targets.
Additionally, if the dive is not canceled before Hat Kid lands on the ground, she will briefly slide forwards and remain in a laying down position until the Jump or Crouch button is pressed again.
Wall-Run and Wall-Jump
By jumping towards a wall, Hat Kid will run up on it, gaining some vertical distance. She can then jump off of it onto a platform or another wall, where she can gain some more height and jump again. If the walls are parallel, this can be repeated to gain as much height as possible. Hat Kid cannot run or jump off the same wall twice in a row. She must jump off of another wall or touch the ground to jump off of that wall again.
If Hat Kid performs a dive and hits a wall, she will bonk her head and fall to the ground. However, if she has the "No Bonk" badge equipped or cancels the dive before hitting the wall, she will start a wall run instead.
If Hat Kid does a double jump onto a wall, she will wall climb higher than if she does a single jump.
Melee
At the start of the game, Hat Kid will be unarmed, hitting anything will cause her to hurt her hand and flinch in pain. This also does no damage. Fortunately, during the first act in Mafia Town, Hat Kid will acquire an umbrella as her primary weapon - which can be swung by pressing the Meelee button. Hat Kid can string together a maximum of 3 swings with the umbrella at once, with a brief waiting period until she can swing again. With the Projectile Badge equipped however, she can only swing twice.
Homing Attack
When airborne above enemies, a button prompt will appear over their heads, indicating that Hat Kid can perform a homing attack. This allows her to quickly lunge herself at the enemy to deal damage and bounce off of them. This is useful for both combat and platforming.
Intermediate Abilities
Although not taught at any point in the game, there are more to Hat Kid's movepool. Those moves aren't necessarily difficult to perform, but they can be immensely useful in both casual platforming and speedrunning.
Dive Boost
If Hat Kid's dive is canceled right after she touches the ground, she will lunge forward, performing a front flip and gaining considerable speed. Doing this in rapid succession can outspeed the Sprint Hat. This is especially useful in Vanessa's Curse.
Momentum Conservation
If Hat Kid performs a Double Jump, touches a balloon / sandbag, or cancel a dive on the ground while having considerable speed, the momentum will be retained, giving her even more speed. This can be chained. For example, if Hat Kid Dive-Boosts into a balloon, then Double-Jump, she can cover the distance of roughly 3.5 Dive-Boosts (This can be done with the red balloon on the path of the rolling-meatclub-Mafia Goon in Mafia Town). This is, While not necessarily a move Hat Kid can perform, an essential part of her other abilities.
Sprint-Jump
With the Sprint Hat equipped, Hat Kid can perform a long jump if the Jump button is pressed while she is sprinting. A Sprint-Jump covers double the distance a normal jump can, about as much as a double jump. However, Hat Kid cannot perform a Double-Jump after a Sprint-Jump.
Sprint-Double-Jump
Commonly abbreviated as "SDJ". If the hat ability button is released right before Hat Kid performs a Sprint-Jump, she can retain the speed and momentum that a Sprint-Jump gives, AND the ability to Double-Jump. Because Double-Jumping retains momentum, she will cover a very large distance, about double that of a regular Sprint-Jump.
Double-Sprint-Jump
Commonly abbreviated as "DSJ". If the player runs towards the edge of the platform they are currently standing on, they can press the jump button twice in quick succession whilst holding the hat ability button to perform a Double-Jump while retaining the speed and momentum that a Sprint-Jump gives. This allows for greater height than a regular Sprint-Jump provides.
Slope Slide
If Hat Kid dives down into a slope, and the slope is steep enough, she will slide down on it, gaining considerable speed. Doing this with a long enough slope will also award her with the Slip-n-Slide achievement.
Slope Slide Boost
Because canceling a dive retains momentum, if Hat Kid cancels her dive while performing a Slope Slide, she will lunge forward with the speed gained from the slide. The distance this move can cover depends on the slope's angle, its length, and how fast Hat Kid is sliding. Although similar, this is different to the Dive Boost, as it can be performed at any point while sliding. This move can help Hat Kid win against the Rocket Mafia in Cheating The Race without the use of the Time-Stop Hat.
Stomp
If Hat Kid lands on her feet on top of an enemy, she will damage that enemy and bounce up a bit. Pretty straightforward.
Cannon-egg Bounce
Similarly to stomping, if Hat Kid lands on top of an egg projectile launched from the cannons in Train Rush or The Finale, she will bounce off of them. Unfortunately, this can only be performed on small egg projectiles, and not the giant ones.
Advanced Abilities
Those are moves primarily utilized in speedrunning play. In other words, check out Speedrun Tech.