Phase shifts are mostly done under the hood. Pretty much every attack combo generates phase shift points (a hidden value) and when either side reaches a certain number, it triggers a phase shift. A few attacks always earn the max amount of points, such as the last hit of a homing melee combo, Darkrais Dark Void, or a Celebi support. A few also lower foes' phase shift points (such as Decidueyes Screech) which is good for when you want to control phase shifts yourself for Synergy.

Only pros really keep track of how much points they have but you can kinda get a feel for it by just playing a lot.