I use all 3 Elements, depending on the Build, and the Target.
Fire: Costs the least magicka, and it's secondary effect is more damage. As it's Damage over Time, you can hit them over, and over to stack it up, and kill them faster. Altercasting (What i call it) or charging one spell while you release another is extremely effective with Firebolts, because it's literally Rapid Fire. I use this more than Impact to finish fights quickly without running myself dry.
Frost: Drains Stamina as a secondary effect, so not only can't Warriors Power Attack, but Trolls (Even Frost Trolls, which are resistant) can't attack at all. Also, Ice Storm can hit the most mobs per casting, so you can spread the love furster, and get decent Magicka efficiency with enough enemies. All Frost also has a Tertiary effect, which Slows enemies so they're easier to hit, and stay away from. Wall of Frost is the best way I know to defeat Dragons once they're grounded, because you can hose their entire footprint, and they can't take off 9Even Frost Dragons) without Stamina. You have 2 hands, so you can also double your DPS with Fire, or Shock Walls.
Ligtning Drains Stamina, and is a Hit-Scan, meaning it doesn't have a travel lag, but draws a line from you hand, and hits whatever's in it's way instantly. This means you don't have to lead moving targets. It's the rarest Resistance in the game, so if you want a beat-all element, this comes closest. With Disintegrate, you lower the target's Resistance, Health, and Armor, which is great to combo with Critical Charge, or other attacks. And finally Lightning Storm is the most powerful spell in the game, if you have the time, and Magicka to cast it.
As for Secondary Schools:
Conjuration takes over when Destruction reaches it's low damage cap (Compared to weapons.) Flaming Familiar is a complete replacement for Fireball, it's cheaper, self guided, and has a duration so you can recover the magicka from casting it. With it's the nastiest combo I know in pure spells. Summons take the heat off of you by tanking attackers, the Flame Atronach kites enemies around, and you can even Lightning Bolt her to explode (With a brief Lag.)
Alteration is adequate defense (If you don't mistake yourself for a tank and charge in like a berserker.) It's also got a lot of utility spells that help out. If you're looking for just 1 other school, this is probably it. Detect Life is nice to see the Sabercats before they Ambush you from behind, and Paralysis can give you a breather, or incapacitate enemies while you hose them down with Wall of Flames.
Illusion is fun for messing with people's heads. I have
one that just uses this with Conjuration to get them fighting each other, send in summons to pound them, and break out the popcorn from the safety of Invisibility.
Restoration provides healing, and Wards to block Dragon Breath. Great perks for managing your stat recovery too (Magicka, Stamina, and Health.)
Hope that helps.