I think you should do BOTH. Dual-cast Destruction until you are half health or lower, then start using resto. Also, if you plan on just using magic, you may want to invest in other skill trees aswell as Destruction and Restoration., such as Alteration.
Both. And find some warrior follover, focus also on some other schools, like Conjuration, or Illusion, or maybe Alteration. And also do a lot of Enchanting.
Unless you're familiar with the game and/or play on lower difficulties, ignoring health may lead to problems, in particular your character's bloodied, twitching corpse
Get at least 250 HP but as a mage I don't think you need more then that. If you don't play at Legendary that is...
You should be more than ok with those 2. Personally, I set my dual casting requirements to level 60. There is no way a newb mage has the mentality or the self control over dual casting spells. By the time you are level 60 in one of those schools you should have the magicka required and the discipline of your casting (you have better understanding of how much magicka you will consume in a fight, making it easy to manage). That is because you will likely get trigger happy and spam dual casting quickly and realize you run out of magicka fast often. Though it does a bit less damage I would avoid dual casting, for now as it is far more efficient to use spells in each individual hand. You can upgrade magicka only and forget about health but only if you keep your distance. If you are playing a mage with robes, then I would suggest investing in flesh spells. Those will save you if you have low HP.
Dual casting destruction is less effective than simply casting the spell from both hands unless you're going to be 1-HKing your enemies with the dual cast. It takes more magicka than its worth in damage to cast a dual casted spell. I've tried it, but I find casting from both hands without dual casting is much more effective. Same goes for restoration. You heal more for less magicka. I honestly don't know why dual casting is even in the game except to look cool and for the stagger effect if you spend another perk on Impact in destruction.
I'm on my first playthrough too, so my advice may not count as much as the more experienced players. My mage is now somewhere around level 24, and heavily focused on destruction magic with some conjuration thrown in. I use dual casting, and have the stagger perk, and have found it very useful. Due to the stagger effect, I can sling fire, lighting or ice bolts at an enemy and stop them advancing at me, which is nice, considering I'm a bit squishy in my robes.
I've got a bit of skill in single weapon too, for those emergencies when I'm out of magica, low on potions, with a melee enemy bearing down on me. Those emergencies have come less and less often though, and I find myself using magic almost exclusively now.