I'm guessing that I'll need to focus on 2 of the 3, but I wanted opinions on which seems best overall for a mage.

Shock damage definitely. Dragons are resistant against frost and fire, undead don't seem to be bothered much by frost, and there are no racial resists against shock damage. Hell, your in the home province of the Nords, all of which have a 50% resist to frost, so go ahead and through that one right out the window.

All of them? Start with electricity though... Electricity = Most effective because it drains Magicka(or stamina, i dont remember) from the enemies and robots are sensetive to it!
Ice = the coolest, no doubt!
Fire = All mages need fire spells, i mean.. Come on.. It's a must!

Shock is the best stat wise. But I'd go for fire, 'cause hell, it's fire.

If I had to choose one, I'd choose Electrical. Nords resist Frost 50%, Dunmer resist Fire 50%. No one resists electrical naturally.

All of the above.

Fire, pow.

depends what you want out of it. if you look at them all have secondary effects, Fire is pure damage, frost anti fighter (since it drains stamina), lightning anti mage (since it drains magica)
I personally did fire for the extra damage some fire immunity is annoying but switching to another damage group to deal with one foe then trudge on isn't a problem.

I am playing a pure mage and originally planned to use: frost (against warriors, stamina drain) and shock (against mages, magicka drain). Shock has definitely become my go-to attack. When out wandering between cities, a lot of creatures are resistant to frost: ice wolves, (ice) bears, frostbite spiders etc. Makes me hesitant to use frost because I assume it will be resisted. Fire is handy because you end up seeing a lot of Draugr, I think a fair few creatures are weak to it as well.

I'm about to make a Death Knight type spellsword with swords and frost-destruction magic only.

I started out as a frost mage, but since SOOO many enemies are resistant to it I swtiched to purely electrical. It works amazingly for me so far. With Destruction dual-casting and the Impact perk, Lightning Bolt has been my best friend in every situation. And now I have both Augmented Lightning perks and the Disintegrate perk. Disintegrate is the most useful elemental perk IMO. Just instant death for something that's down to low HP.