Mages are extreme elemental damage-dealers. They can use 4 types of magic. Fire, Ice, Lightning and Mana magic. All types of magic have single strike and full party elemental attacks. All elemental magic also has veils, except for Mana. They also have Space magic. Space magic has Portal Jump (ポータルジャンプ) and Exit (イグジット) that can be useful when you explore. They also have Pharmaceuticals (薬学) and Discerning Judgement (鑑定学). If you have SP left, you can level up these skills. Mages have low DEF and are MANA intense, so they will need extra protection and MANA recovery, though.
Fire has a big disadvantage compared to the other elemental magic. Many bosses have resistance to fire and not many mobs are weak agains fire. Fire has a great full party attack, though.
Many dragons are weak against ice, but many mobs have resistance to Ice.
Lightning is a good pick. Many dragons and mobs are weak against lightning, so lightning is a pretty safe pick.
Mana is the safest pick. Mana magic ignores both resistance and weakness, so this the most pretty neutral. Mana magic has Healing Mana (ヒーリングマナ) and the awesome Mage's Concentrate (メイジズコンセント). Mana has a disadvantage, though. Mana is a little weaker than the other elemental attacks and Mana mages don't have veils. Mana is the easiest way to get Space Magics EX spell, End of the World (ジエンドオブワールド) that has a chance of inflicting skill-seal.
When you have all skills you need, max INT bonus. You might want a Healer that can use Mana Craft or a Princess that can use Song of the Moonlight. There are many types of Mages:
Single Elemental and Mana mage picks one elemental + Mana magic. This Mage have Veil + Mage's Concentrate + single target elemental/Mana attack. You'll be using elemental hit-all spell on mobs, if they have resistance to that element use Heaven's Press (ヘヴンズプレス) Against bosses/dragons use Mage's Concentrate + single target elemental spell. If the boss/dragons are resistant to that element use Mana Bullet instead. If you want you can pick up elemental EX spell too.
Pure Mana type is a Mage that only uses Mana magic. Heaven's Press for random encounter and Mage's Concentrate → Mana Bullet for bosses/dragons. You can raise Healing Mana if want, but, 3 MANA is pretty small so it's not necessary. When you're have all skills + INT bonus, go for Planet Mastery (ポータルジャンプ) or Pharmaceuticals or Discerning Judgement. Late game, You can get End of the World, but, keep in mind that Heaven's Press and Mana Bullet is required at level 5 and cost 7 MANA.
Tri Elemental is a Mage that uses Fire, Ice and Lightning. This Mage is one of the most SP intense Mage (together with Abjuration Mage). This Mage has the advantage to deal with both weakness and resistance. You won't get Mage's Concentrate before late game (If you haven't raised your skills to max) but that shoudn't be a big problem if a foe has a elemental weakness.
The Abjuration Mage is abnormal. The Mage sits on the front row and cast a veil on himself and then he cast a maxed Craft Shield (シールドクラフト) and continue recast it each turn. Though, placing a Mage at the front row is a fast ticket to die. So you might want a Knight with Defence Shift, a Fighter with the EX skill Maxing Call or a Princess with Solid Rhyme. Also, you must have a Healer with Craft Mana or a Princess with Song of the Moonlight because recasting Craft Shield every turn is very MANA intense. This Mage is also SP intense, so this type isn't available in early game.