The most important thing you should know about battles is that the colour of the gems you link are connected to each of your heroes.
With a full team of four heroes, each will correspond to a type of gem you'll be linking. Linking their gems will power them up, allow them to use star skills, and of course, attack.
This also means that linking the purple arrow rank gems won't do damage, and knocked out or stunned characters won't be attacking, and their gems should wait.
Though, that certainly isn't to say you should ignore the rank gems entirely - just when you need immediate damage, they're not the first choice.
But if you're in this for the long haul, then you should use the rank gems early on and in big combos, to fill your rank meter.
Ranking up in battle will allow you to add a damage multiplier and other bonuses, which, of course, means dealing more damage. Hell yeah.
As you link gems and do battle you will fill the mana bars which lay just underneath the character icons, and once it reads "MAX" you can tap the character icon to let a large scale attack.
You should pick your moment when to use these though. Some give a single enemy a large amount of damage or multiple hits, and others will attack multiple.
Identifying the effects of the attacks by looking at each character in the party menu, and then using them when it most effective, will be key to winning battles.
Back from the dead
It can happen, it's not your fault, you just have to move on. If one of your characters go down in battle, it can be awkward.
But it's okay, they can be brought back with a fair bit of effort. Linking the gems of their colour after they die won't do damage to your enemies, but it will, eventually, bring them back.
Having them back in action is of course for the best, but if you're nearing victory and just need a bit more damage done, maybe they can wait.