Controller - struggling with targeting

[Derio]

I have mine set to where I only target NPCs when i have my weapon sheathed. However I would not recommend that if you are a healer.

[EusisLandale]

Assuming you have access to a keyboard as well, F1-8 can be used to quickly target a specific party member.
L1+R3 switches the controller to a virtual mouse and back, with L2/R2 functioning as left and right mouse buttons. Depending on how quickly you can do this it may help as well.
The PS4 controller's touch pad also functions as a mouse, though far less precisely than the virtual mouse mode; Drag to move the cursor, tap to click, hold a double tap to hold the button. That last command in particular is especially touchy about working right, but this method may also help

[Rowde]

You can also adjust what is targetable in Character Configuration.

Character Configuration > Control Settings > Filters

You can adjust settings for the weapon sheathed and weapon drawn states AND you can enable additional filters activated by pressing L1 + "a face button"

[LineageRazor]

There is also an option in the settings that allows you to never target other players' pets. This is pretty useful - except in extremely rare cases when you need to be able to do so. For example, a dungeon run where your tank d/c'd and the group decides to continue using the Summoner's Titan-Egi to tank. (I've been in this boat many times, both as healer and as Summoner!) Just turn the option off for those rare cases.