Some parts of it remain awkward, but there are some things that can help:

You can set up multiple targeting filters. This can help in situations like FATEs where there might be several enemies, several players, and several NPCs all clustered together, and you want to focus on a particular group. To do this, go into Character Options and find the targeting filters tab. There's two sections: at the top it lets you set filters for what to target based on whether your weapon is drawn or sheathed and below that it lets you set up to four custom filters. You should only enable one of the two sections or they'll interfere with each other. Personally, I prefer the bottom option of the four custom filters. That lets me set it so that L1+[square] will put me into "Enemies" targeting mode (useful for both DPS and tanks), L1+[circle] puts me into "Friends" targeting mode (useful for healing), and L1+[triangle] puts me into "All" targeting mode (which I like when running around town or dealing with NPCs). I've occasionally used L1+[X] for custom filters like selecting just enemies that have already been marked with one of those 1,2,3.. priority indicators. Once you have those filters set up, L1 + a face button changes your targeting mode, and then left/right buttons alternate between just that type of target until you change your mode to something else. It doesn't help the odd order in which enemies are targeted, but at least you aren't scrolling through several players, NPCs, companions, pets, and minions, when you just want to target your next enemy.

Left-right buttons will soft-target scroll through whichever types of targets are included in your current filter (described above), although in a seemingly random order. (L2 or R2) + (L1 or R1) will hard-target scroll through any enemies in the area, though that's also in a random order. Up-down buttons will soft-target scroll through the players in your party, in the order they appear in the party list. L1 + up-down buttons will soft-target scroll through the enemies you've already aggroed in the order they appear in your enemy list. At this point it's worth describing the distinction between soft and hard targets:

If you cycle your target to the ladybug in front of you and press X, it becomes a hard target and will be the target of any attacks you cast until you select something else. If, while you have that ladybug hard-targeted, you press the right directional button to soft target the squirrel next to it (but don't press X, so it remains soft rather than hard targeted) and fire off a couple attacks, the first of those attacks will hit the squirrel (your soft target), but then the targeting automatically switches back to your hard target (the ladybug) which gets hit by the second of those attacks. This double targeting pattern is particularly useful for healers, who often want a particular target (the tank) to be the focus of the majority of their actions, but still have the flexibility to fire off single heals or attacks to other players or enemies. By hard-targeting the tank and only soft-targeting anything else, they quickly (and automatically) get back to the tank, who's their most important and common target. Tanks and damage dealers can also make use of it if they want to focus the majority of their attacks on one particular enemy, but then fire off an occasional support skill to someone else without losing their main target. (When you don't want two different targets selected that way, just get into the habit of pressing X after soft-targeting something, as that turns your current soft-target into your new hard-target.)