I have a basic understanding of combat that let me progress through the entire game so far, here's what I understand:
You have to distribute AP each turn between stats to increase your DEX (hit chance), STR (damage inflicted on hit), AGI (chance to dodge incoming attack completely) and DEF (damage reduced upon taking a hit)
DEX and AGI work in % chance
STR and DEF work in flat amounts of damage
Usually it's worth playing around with the values before locking them in so that you can find the most efficient use, because the increase are not linear, sometimes you will find situations like (I am making up a example):
0 DEX -> 49% chance to hit
1 DEX -> 50% chance to hit
2 DEX -> 71% chance to hit
3 DEX -> 72% chance to hit
You can also use your turn + AP to perform certain special action like healing with "Recover"
Regarding AP refilling, from my understanding they refill *AFTER* the stated amount of turns has elapsed. For example, if it says 5 turns for AP recharge, I've noticed you will not get AP back on turn 5 but instead at the beginning of turn 6, then 11 etc..
Finally, gear and level play an importance although the progression is fairly linear.
