Each character is assigned with one of five elements: Heart, Skill, Body, Bravery or Wisdom. Match up the elemental affinities properly before get into battle.
Here are the correlations between elements: Heart is effective against Skill but not against Body; Skill is effective against Body; Body is effective against Heart; Bravery and Wisdom types deal a lot damage against each other.
Keep in mind Wider-ranged characters can attack more enemies at once.
Finish Emergency Missions to get boss character to join your team.
Use Field Skills with allies nearby for a more effective Combination Attack.