Run/Dash
To run forward, press 656 in quick succession when on the ground, and keep holding 6 to continue running forward. You can cancel the run into attacks and Barrier Guard. If you stop running and don't cancel into anything, you will slide to a stop. This slide animation is completely vulnerable.
Note that some characters like Carl and Hazama perform a short step forward instead of a run. This means that you will always move forward a set distance unlike a typical run.
Run/Dash Brake
Because runs typically have a vulnerable period after they are done, players typically cancel the run into Barrier Guard briefly, then continue doing whatever they wanted to do. This is a very powerful technique to maintain a specific range vs the opponent, and you should be in the habit of doing it everytime you want to stop running.
Momentum When Running
When performing attacks and jumps while running, some of the momentum is carried over to your next action. For example, running forward then jumping forward makes jump much farther than if you were to do a normal forward jump. If you do an attack while running, your character will slide a bit while performing an attack. This property is used often in combos to give them more reach and in footsies for the same reason.
Backstep
To perform a backstep, press 454 quickly. Most backsteps have a bit of invulnerability at the start, so they are useful for escaping enemy attacks if timed correctly.
Jump
To jump, hold the joystick in any upward direction. In general, when you jump, you will go through a jump start-up phase and then leap up. While in jump start-up, your hitbox is considered grounded, you are invulnerable to ground throws, and it's possible to cancel into things like special attacks or gold burst.
High Jump
To high jump (also known as super jump), start from neutral, then quickly press any downward direction, then any upward direction. High Jumps have startup just like normal jumps.
Double Jump
To double jump, press any upward direction, when in the air. You can not perform a double jump if you already performed an air dash. Taokaka can triple jump, and Tager can not double jump at all (poor Tager).
Air Dash
To air dash, press 656 or 454 twice in quick succession when in the air. Bang and Taokaka can double air dash, and Tager can not air dash at all.
Instant Air Dash
This is a technique done that allows people to perform an air dash immediately after jumping. To instant air dash press 956 to IAD forward, press 754 to IAD backward.
Air Movement Rules
Whenever you go into the air you can do either an air dash or a double jump, but not both. Being able to do either command is called having an "air option". You do not regain you air options until you touch the ground again.