For players in your party, up/down on directional pad.

To reach players not in your party (or NPCs you need to heal for some of the solo instances), set up filtering in character settings:
Go to the target filtering tab in Character Settings. It has two sections. At the top it has an option for switching target filtering based on whether your weapon is drawn or not, and below that an option for switching it based on up to four filters you select. You should only have one of those two types enabled. (If you enable both, they'll interfere with each other.) I disable the weapon drawn/sheathed part and enable the four manual filters in the bottom section. I have it set so that L1 + square sets me into "Enemies" targeting mode, L1 + circle sets me to "Friends" targeting mode and L1 + triangle sets me to "All" targeting. (I've occasionally also used L1 + X for various "Custom" targeting modes.)

With this pattern set up, L1 + a face button determines your targeting mode, and then left and right on the directional buttons will only cycle between those types of targets until you switch your targeting to a different mode. (Unfortunately, it doesn't fix the fact that which friend or enemy is first or next on its list seems random rather than based on proximity or direction.)

Note: this works well if your controller is set so you hold R2 and/or L2 while triggering actions. I've heard it gets more awkward if you use toggle or mixed mode, but haven't tried those myself to know another good workaround for that. I just think of this as a good reason for favoring the "hold" method over "toggle" or "mixed".