Jump: Swipe in any direction. The angle you swipe and the distance will affect the horizontal and vertical speeds.
Double jump: Swipe again in the air. You can only perform one double jump after you leave the ground. To get another double jump, simply land again. Swiping downwards won't consume the double jump.
Air Movement: When you no longer have another double jump, swipe left, right or down to change directions in the air.