Leaving aside the issue of varying resistances that you'll encounter:
As a rule of thumb, Fire spells will do the most damage, including a chance to cause enemies to catch fire and continue to take damage for a few seconds. Frost spells drain opponents stamina, and slow their movement. Shock spells drain Magicka, and can chain to other, nearby opponents.
In addition, there are several perks specific to each damage type. Above and beyond the pure damage increase perks, the Disintegrate, Deep Freeze, and Intense Flames perks add a chance for an additional effect when casting Shock, Frost, or Fire spells on a target, respectively.
I would work on restoration to 60, to provide additional magicka regen, and cheap Close Wounds. The spells and perks beyond 60 are very much optional.
Alteration is indeed useful, as without damage reduction you will be very squishy at later levels. The entire tree is pretty good, with very useful perks - but it does need a lot of them.
Conjuration is useful as a means of providing a wall between you and enemies. However, it may not be to your playstyle and you can certainly do just fine without it.
Illusion is good. Invisibility is not perfect, and you can still be detected but you can use it to infiltrate at a short to medium range. The CC spells of Illusion require high investment in the relevant perks to remain effective end-game, but you can focus on a particular effect (e.g. calm) to save a few perk points. On the other hand, Aspect Of Terror also boosts your destruction fire damage by 10 points.
I would consider alchemy too, as one of the few ways to reliably boost destruction damage. (glowing mushrooms with nightshade are probably the most common ingredients that offer the effect).
Enchant reduced cost destruction equipment, and use remaining slots to boost magicka regen for use on the other schools. Magic resistance is useful, but with Alteration perks and any base Racial trait (are you a Breton?) one piece should be fine.
As mentioned by others, a character planner can help ensure you have enough perk points. Magic can be perk intensive!
EDIT:
Some choices, in response to a comment:
1) A perk planner spreadsheet by Shiloh hosted at SkyrimNexus
2) Graphical Perk Planner by PsychoHampster hosted at SkyrimNexus
3) There are also some excellent online ones, I believe one by IGN as well. But I use offline ones so I don't have links.
Best Answer
Making it "easier to cast" is relatively straightforward - it reduces the Magicka cost of the spell, meaning you can cast them more often.
This is similar to how things worked in Oblivion. (I'd link you to the UESP page documenting this, but the Site's still getting hit pretty hard from all the Skyrim traffic)
Each spell has a "base cost", which is modified by the following formula:
This means that at 100 skill, spells cost 60% of the base cast (further modified by perks and equipment).
This isn't combat related, but as Affe has pointed out below, increasing your magic skills is required to buy high-end spells from vendors.