Mages have access to three kinds of elemental magic: Fire, Frost and Lightning.
Main spells can be learned in the Talent Tree tab of the Talents interface (hotkey N). Additional spells and improvements of them can be found in the Talent Grids when the mage reaches level 10.
Talent Grids allow additional customization of a mage build. Some of these talents can only be found on one of the grids, while others are shared by all three of them.