I have started a mage character, High Elf, mage stone, hasn't swung a sword yet. These are things I've learned, as a stealthy fighter playing a mage.
Runes are effectivly stored magic. Stick them on walls and let your personal magic pool regenerate for the fight. Then, in the fight you can use the rune to delay hostiles when you retreat to regenerate.
As a warrior, your Stamina is important. As a mage, your Magicka is even more important. If you choose to make a specialized mage, don't bother putting a point into Stamina, and you can get away with neglecting your Health a bit too. You never have enough Magicka.
Necromancer Robes are easy to come by, and they have 25% more magic regen than Novice Robes.
Mages are lighter than the lightest Theif. You should know how your enemies move, so you can dodge them.
Remember how when you were a warrior, that one mage would always backpedal away from you shooting magic? Now you're that mage.
Since you are not wearing any armor, you can sprint for a long time. Take advantage of that.
Your weapon and shield slow you down. Your hands don't slow you down, even if you're holding fire.
Fire is good against Draugr, but they resist cold.
Shock is not resisted by anything but a Storm Atronoch, but it costs more Magic than other spells. It also goes really fast and is good for ranged duels with Mages.
Cold is effective against warriors, but since the majority of enemies are either Nords or Draugr that resist cold, it gets nerfed pretty bad.
Fire is good against about everything, but the capstone perk where things run away when they are damaged would get annoying fast.
A mage does not have to use Destruction spells unless his enemies are Automatons or they are Undead. Mages can use Frenzy varients to make enemies kill each other and then they can raise the dead to fight on.
More often than not, your enemy's outline will be blurred by flames or electrical arcs. If you rely on visual cues for when to dodge, that may mess you up. Listening becomes more important.
I'll come up with more later, as I continue playing. Please add to this list.