I play on PS4, too, and I find this helps for me:
Under Character Configuration, Filters tab, uncheck "Enable Target Cycling" (if it's set), and instead check "Enable Filter Customization".
Under that are four sections of targeting filters you can swap between. I find it good to have one of them set to Friends and another to Enemies. You can customize the other two however you want. (Mine are Friends, Enemies, All, and a custom one where I checked only the marked targets box.)
Once that's set up, you can adjust your targeting controls using L1 + a face button. In my case for instance, L1 + [square] will set me to "Friends" targeting mode, while L1 + [circle] will set me to "Enemies" targeting mode. These targeting modes control the way the right/left buttons work for selecting targets. If you're in a quest that needs you to heal an NPC, or at a FATE where another player (not in your party) needs a heal or raise, switch to "Friends" targeting mode and your right/left buttons will now skip over all the mobs in the area and only target players and friendly NPCs. Then when you want to attack, switch to "Enemies" targeting mode using L1 + whichever face button you assigned it to. Now the left/right buttons will cycle only between enemies, skipping over players and NPCs.
The reason I specified Friends mode being for healing those not in your party is because you have another option for targeting party members. The up/down controller buttons cycle between party members in the order they appear on your party list. Since that maintains a consistent order independent of where they're at at the time, and since it displays all their health bars and status effects right there, it's the easier way to select the party member currently most in need of your aid.
Similar to up/down for the party list, L1 + up/down cycles through the enemy list once you have aggro on enemies. It's not quite as reliable as the party list, since enemies don't appear there until after you have at least some aggro. (An enemy can arrive and start attacking one of your teammates, but until you attack it or heal the member that it's attacking, it might not notice you in order to show up on your enemy list.) But it's frequently the easiest way to target the mob that's currently taking the most damage, or for tanks to target the one they just lost aggro on.
(L1 + right/left cycles through the alliance list, but you won't get to full sized raids that actually have an alliance of parties until at least level 50, and it's optional content then, so don't worry about it for now.)