Like Rangers_Fureva said, R2 is what you probably want to look into... R2 will shuffle you forward while facing the ball (meaning that if you move perpendicular to the ball you'll strafe). You can do this at any time in the game to break the 'script' that your player is in and start shuffling; it's really useful in the box to get ahead of defenders to receive crosses, or anytime the ball is in the air, to establish a spot under it.
R1 and R2 together will also break any scripts, but it will also free up your movement entirely, unlike R2 which makes you face the ball. Doing the R1+R2 move (super cancel) is probably best for chasing down passes because it lets you move faster... Just make sure you direct your running well because your guy won't run straight at the ball automatically like usual.
I like to use super cancel in the box sometimes too on crosses, right when the ball's about to come, sometimes you can charge right by defenders for a head with a nice bit of momentum on it.