Here is a run down of how the combat works (I played the game for 20 hours at this point, so I think I can help with this

).
The Burst meter determines the power-level of your 'main' attacks; the higher it is, the stronger your hits will hit and vice versa if the meter lowers. You can make it rise again if you switch with your party member, with them usually using a sword skill to buy your time for your gauge to fill up.
The square attack is the 'stun' attacks; once you press it, most of the time you will stun your foe for a bit (when your party member tells you to do it, if you pull it off, they use a strong sword skill as a follow up). You also have your 'block' skill mapped to triangle. If you pull it off at the right time, you get 100 SP points and they suffer from what looks like a simple 'stun' state. Both operate on a cool-down meter though, so type your usage of them well.
The way to manage your SP is to consonantly praise your partner. When they use a sword skill, follow your commands (telling them to heal, go in DEF mode, use skill, ect) and once your praise them, you get 50 SP points. You also get them through crystals (which offer 100-200 SP points per use) or potions (which enable the auto-refill of your SP to slowly to greatly increase).
Speaking of the sword skills, you have your attack ones. Depending on the weapon type you chose to have (Dual Swords, One Sword, Two-Handed Sword, Axe, Mace, Spear, Lance, ect), you have different attacks to use. Considering the anime/light book novels, you have the most Dual Sword/One Sword Skills right out of the gate, so don't worry about not being able to fight well

. The only thing to keep in mind though is two things; your skill tree and your SP management. The former isn't that much of a worry, but it is important to carefully decide what skills you want based on your skill points (you get a lot of them if you level up enough, so don't worry

). The latter is something to worry about (as you attack Sword Skills eat up 100-150 SP points), but there are ways to deal with this (mentioned latter in this post).
You also have to manage your buffs/de-buffs as you fight. You have a verity of battle skills you can use (giving you a EXP boost for a while, increasing your hit chance, you power, you HP regeneration speed, ect). Some of them use SP while others operate on a cool-down set up. You can also buff your party with potions you get from shops and quests.
Using your partner is also important as you can tell them to heal, use DEF stance, use there skills and more. This even extends to the 'MP' Mode and Boss rooms, where your commands are followed by everyone, not just your party. Using this right can really save your party from dying.
I know this is a bit exhaustive XD, so I hope this helps

.