To cast non-combat spells while adventuring, select the character you'd like to cast a spell from the drop down list in the upper left panel, and then hit the 'Cast Spell' button. Provided the character you select knows at least one non-combat spell and has enough spell points to cast it, you will be asked to select a spell to cast, and then, depending on the nature of the spell, on whom. Once the spell is cast, the character will lose however many spell points were required to cast the chosen spell. This number is given in parentheses next to each spell in the drop down list. Combat-specific spells can obviously only be cast when in combat. See the Combat help page for more information.