Okay, movement in Smash Bros is a tricky topic, there’s a lot of nuances to it that don’t exist in any other game I know of, so lets get down to business.
Up above is an image (made by Kadano) detailing all of the actions possible from a neutral standing state using the control stick alone, assuming you are facing right. When you tilt the stick forward, you will begin to walk forwards, with speed proportional to how far forwards the stick is pressed. If the stick is moved into the dash region within 2 frames of it being in the dead zone, it will initiate the initial dash animation, if after 2 frames it is not in the dash region, a dash cannot be initiated regardless of the movement of the stick until it enters the dead zone again. If the stick is moved back into the tilt turn or smash turn areas, then the character will initiate the turning animation, if it is in the smash turn area on the first frame of the turn animation, then the initial dash animation will be started in that direction.
While walking, you are allowed to perform any standing action, forward tilt, down tilt, up tilt, fsmash, down smash, up smash, and all your B moves. This means walking is a nice and delicate way to move while being capable of performing any attack at any time.
As noted above, when you “smash” forward, you’ll enter an animation state called initial dash. This animation plays for a different number of frames per-character, and if the stick is held forwards, it will transition into the run animation when it ends. If the stick is released, the initial dash animation will continue to play until it concludes, but you will not maintain your speed forwards, with friction slowing you to a halt (this is called a fox trot).
During the initial dash animation, you are not allowed to crouch, attempting to do so will make you crouch after a fox trot. You cannot perform any tilt or smash attacks during the initial dash animation, except for up smash (because jump cancels into up smash, and you are allowed to jump from all grounded non-attacking states.). If you smash turn during the initial dash state, you will re-enter the initial dash state in the opposite direction. There is a 2 frame leniency for this (unlike the 1 frame leniency for trying to dash in the opposite animation from standing), not reaching the opposite side fast enough will result in a fox trot.
This means that you can rapidly alternate directions to begin the initial dash animation over and over again in the opposite direction. This is called dash dancing. If you allow the initial dash animation to play out completely, entering a run, attempting to turn back will result in a long turnaround animation playing where you lose a lot of your speed and cannot perform any action except jump. Jumping during this animation will orient you in your original facing direction in Melee, and the opposite facing direction in Project M (called reverse aerial rush).
Because the turnaround animation during run is so long, many players elect to stay in dash dance to get the fastest turn times, allowing them to move with more agility than the run animation over a shorter range. The thing to get familiar with is learning how long you can hold dash in a direction before needing to turn around. This means that there is effectively a distance you’re allowed to move before needing to turn back. When you learn this distance well enough, you’ll be able to move at maximum speed over short distances, weaving in and out of opponent’s attack ranges. It’s possible to run across the stage, staying entirely in initial dash by turning back, then forwards every time you’re about to hit the edge of your range. This is a great way to practice using the dash dance purposefully. Varying your dash lengths and having great precise control over your dash will allow you to whiff punish any move.
During the initial dash animation, the only attacks you’re allowed to perform are the dash attack, grab (the running grab animation is slower, it’s recommended that you jump cancel grab to get your faster standing grab), your B moves, and up smash. However every time you turn during the initial dash animation, there is 1 frame where the character is in a neutral standing state. On this single frame, you are allowed to perform any neutral standing option, assuming you have good enough timing and dexterity. Performing an action on this frame is called a Pivot. Pivots are extremely tricky, but allow you to move at maximum speed and attack with impunity. Because they’re so difficult, many people only use pivots for specific applications, like moving in for a quick smash attack.
Once you enter the run state after the initial dash, your options increase a little, because you are allowed to cancel run with crouch at any time, and perform any move you normally can out of crouch (all special attacks, all smash attacks, all tilts). Worth noting is that crouching, then dashing the opposite direction is faster in a run than trying to do the run turnaround. This is called a Cactaur Dash.
Wavedashing is a technique performed by airdodging at an angle into the ground as soon as you leave the ground from jumpsquat. When you hit the ground, there are 10 frames where you cannot act due to landing lag. This means wavedashes effectively have a startup time of 10 frames + your character’s jumpsquat, assuming you do the wavedash frameperfect. You can also do this as you land from a jump or come up through a platform (fastest way to land on most platforms), incurring the same 10 frames of landing lag. The wavedashes of most characters are slower than dashing, with the exception of the characters with the absolute best wavedashes, like Luigi and Ice Climbers. Wavedashes are nice because they allow you to move at dash-like speeds without committing to the more limited set of dash options, as well as retain the same facing direction. So you can wavedash backwards while facing forwards. They help fill in a few holes in most characters’ mobility options. Wavedashes are bad, because they have a longer startup time than dashes and walking. During the jumpsquat, the wavedash inherits whatever momentum you had moving forwards, so dashing into a wavedash will make the wavedash move further.
The angle at which you dodge into the ground also affects wavedash length. More shallow angles that are closer to parallel with the ground will travel further along it. More deep angles that are perpendicular to the ground will travel with less distance. You cannot wavedash perfectly to the left or right, you’ll just get an airdodge, however you can waveland perfectly to the left or right when you jump up through a platform, or land on the ground. You need to do this exactly as you land, close to frame perfect if it isn’t totally frameperfect. Doing this will move you a lot further and faster than a normal wavedash, allowing even characters with terrible wavedashes like Ganondorf to move amazing distances.
If you are facing with your back to a ledge and have momentum, you will slide off the edge. Wavedashes allow you to do this, making them great for grabbing the ledge quickly to edgehog. Sliding off a ledge also can cancel any special animation, allowing you to attack faster, and attack animations can slide off ledges both backwards and forwards. In shield, sliding off ledges will put you into tumble, which can be taken advantage of by opponents.