Others reading there should be no real spoilers you would not find out about in about 5 minutes of playing the game. I probably also got most of the terms wrong and yes it will look very silly out of context.
Short version stick with it- it should all become clear soon enough.
Resonance of fate is probably my favourite 360/console RPG in a long time with the fighting system playing a huge part in it (there are a fair few by the way and it is not as under-served as the original xbox -
http://games.teamxbox.com/index/xbox-360/by-genre/18/ and many are either Western RPGs or Japanese ones with interesting fighting systems (I had a very bad case of early Final Fantasy battle system fatigue coming into this go around), perhaps not as many as the handhelds but still worth a look and I would even go so far as to say unless you are especially fond of Nippon Ichi's stuff- Atelier or Disgaea mainly you are not actually missing out on that much, if anything, really).
Yeah the system is not brilliant until you start completing a few more crystals (make sure you unlock all the levels you are in- there are quite a few in the first couple too) and modding your weapons (or even better dual wielding), also find a shop and buy the auto trigger accessories- they are very useful when you are training up.
The system is two fold- it makes no real sense to anybody that has ever used a weapon but it is great for the game.
You do scratch damage- this comes from machine guns and your fists (do not use your fists). You can do a lot of scratch damage quite easily though.
You do regular damage- this comes from handguns and grenades usually (although some grenades do other things as well). This turns any remaining scratch damage (enemies have a nasty habit of regenerating scratch damage) into full damage and takes off a token sum by itself.
This means the game pretty much boils down to doing a lot of scratch damage and then finishing things off with regular damage.
Each character gets a timed round which you can walk around in or charge your weapon in. In addition to this you can do what I think is called a hero run or something like that- this consumes a crystal charge (which you have precious little of at the start and could always do with more of as you go on) that is available to see at the bottom of the screen (they are coloured red and refill as you destroy enemies and armour of enemies). This move however is quite powerful as it allows you to run, charge (it slows time a slight bit- you can charge more during a run than you can in a normal move) and fire as many times as you against as many enemies as you can charge and fire at before you get to the end of it (there are things you can do like jump which has some advantages- it sprays bullets all over your target which is nice for avoiding armour but it does spray them everywhere).
You have two other party members as well- if your hero run intersects with the line between them (you choose the direction of the run and send them on their way) you get a charge point or something but any non hero run movement or the thing I am about to explain will lose a point (or more if you carry on moving/charging), you can build up charge points but you are a bit limited by amount of crystal charges at first (you also have limits later on as you try to balance things).
Once you have a charge you press the top button and it should connect all your characters in a triangle, pressing the button to launch this causes all three characters to do as many hero runs as you have points for- this is your ultimate attack and done correctly can even wipe out bosses very quickly.
You can change who goes first and which order the moves go in (although you can swap around mid run) though.
After a while this becomes your mainstay battle plan but for the first hour or two you will more or less be hiding behind bunkers and using charge as little as possible (although there still is an element of go big or go home).
Your health is also two fold- you have your main health for each character which when it reduces to zero by way of scratch damage you lose a crystal itself (it fractures and flies all over the battlefield- enemies can be healed by these) along with any charge that might have been in it. You also have enemy attacks (mainly poison related) that reduce your effective maximum health until such time as you get to a healing point or manually sort it.
Secondly if you lose all crystal charges you enter fear mode or something in which case your characters become inaccurate (I think take longer to charge as well), damage to you becomes real and you stand a chance of losing against even basic enemies. I have lost battles I was all but set to win because of this.
After this are are loads of little tweaks and considerations for health, damage and weapons but that becomes clear soon enough. The main thing I want to say is the sub levels as any lost crystals (say if you run from a fight having lost some) and permanent damage carries over between sections.