up/down soft-cycles through your party list
L1 + up/down soft-cycles through your enemy list
left/right soft-cycles through your current filter
Your current filter is configurable. By default, it starts out including everything, enemies, players, objects, NPCs, minions, etc. Under the "Filters" tab of Character Settings, however, there are options to configure it to be more specific. There are a couple ways of doing that. One (at the top of that tab) is to set a list of target types that you'll cycle between when your weapon is drawn and another for what you'll cycle between when your weapon is sheathed. The other (below it) lets you set up to four targeting filters that you can switch between at will by using L1 + a face button (circle, triangle, square, or X). One thing to keep in mind when setting this up, you only want to enable one of the two patterns, either the sheathed/drawn weapon version or the four selectable filters version. If you try to set both up, they'll interfere with each other and become confusing.
Personally, I prefer the later method, as it gives me more control. I use L1+circle to switch to "Enemies" targeting mode, L1+square to switch to "Friends" targeting mode, L1+triangle to switch to "All" targeting mode, and L1+X to switch to a customized filter for specifically targeting marked enemies only. (The name of your currently selected filter is shown above your HP/MP/TP bar, so it's useful to use the named filter patterns for all but one of your filters. If you have different filters each named "Custom" you won't have that visibility to which is currently active.)
As for soft targets versus hard targets, that's actually quite useful, but it might take getting used to. When you cycle through to a new target, it will be a soft target unless you press X (confirm) to make it your hard target. You can use a skill and it will apply to your soft target if you have one, but then your target will change right back to your current hard target. (This is very handy if you want to toss a quick Goad to the tank in the middle of your damage combo, as your target goes right back to the mob you were fighting, or if you want to DoT an enemy or two while keeping the tank as your primary healing target, etc.) When you want to really switch targets, so the new one is your main (hard) target, just remember to press the confirmation X, whereas if you just want to send a single skill to that target and go back to where you were, don't confirm it with X.
That's also useful, particularly for melee since it tends to do a better job of selecting a close-by target. It just gives you less choice for selecting something else when you need to, so it's worth also getting used to the cycling target filters. (The party list up/down and enemy list L1 + up/down are especially useful when there's one specific target you need to hit.)