You can buy Spells in the Store or in some shops at Street View. Every character starts with some base spells. Your base spells generate Mana which will go int the upper-left when in a battle. This Mana is used to cast other, more powerful spells. Look in your Spellbook next to your Profile and read the spells to see what they cost and what they result in. Using the more powerful spells usually takes some planning and "setting up". Also, don't forget to look at your party-member's spells too! You will also get more "Slots" for spells as your level increases and unlock new spells you can buy.