Hello, and welcome!
WARNING: really long post incoming!
You really REALLY don't need to sweat your stat points/leveling gains. The game is highly forgiving. Honestly. If it makes you happy to squeeze the most stats out of every action, go ahead, but I find that adds a lot more stress, and honestly doesn't make that much difference in game play. Leveling up quickly to be able to get good gear from drops is, in my mind, at least as effective, and definitely more fun, than leveling slowly, being stuck in leather armor even though your armor skill is high, just so that you can push around bandits with no health...
Mages can be ridiculously powerful in this game, and it's my favorite style of playing. All the magic schools are well designed and highly useful.
Destruction: for killing, of course. With a enough skill level/magicka/well-designed wide-area spells, you can take down an entire army in a couple of seconds--seriously. Being able to combine weakness to magic, weakness to fire, and a wide-area fire effect in one spell is the reason why spellcasting will always be superior to physical attacks. Of course if you plan to nuke an entire city out of existence, you will need fortify magicka enchantments or potions and some restore magicka potions, but yes you can do it, if you so choose.
Illusion to hide/manipulate/crowd control/avoid bounties. You can charm people into giving you information, you can see in the dark or create light around you, and you can make yourself invisible. Want to murder someone in broad daylight without anybody batting an eyelash? If you cast a frenzy spell on your target, heck, the guards will probably do the dirty work for you! Illusion opens up lots of cat-and-mouse opportunities where you can play with your victims.
Restoration is basically immortality for yourself and your friends as long as you have magicka. Being able to cure diseases and buff stats is just icing. I can't think of any reason why a character of any class wouldn't pick restoration as a major skill.
Alteration is for buffing yourself and others--great when you need to keep weak escorts alive. Once you're a mage's guild apprentice, you can craft a bunch of feather spells and never worry about whether or not you should loot that daedric weapon. Once your alchemy skill gets higher, you might want to use potions for long-term feathering, but alteration lets you carry thousands of pounds even at low levels, if you don't mind recasting the spells all the time.
Mysticism for, well to be honest I use mysticism as marker effects in my spells. I add a second of soul trapping to my destruction nukes to let me know when an enemy dies (you get a message saying there's no soul stone big enough for blah blah) and I add a minimum-range life detect effect to powerful group buffing spells, so that I can use the little blue egg shape to gauge how soon I need to re-buff my friends. Beyond that, I think of mysticism as a way to raise my intelligence once I've maxed out alchemy...
Alchemy is an amazing skill that all character types should master, but IMO none should pick as a main skill. It levels up really fast, and it's useful in any situation. You can use it to make money with zero effort; just buy out an innkeeper's food, make restore fatigue potions, and sell back for a nice profit. It's easy to get your +5 to intelligence by making 10 skill points' worth of potions each time you level up, and all it takes is a trip to the IC market district--clear out the silver carafe, the main ingredient, and the merchant's inn and you should be able to make more than enough potions to get your points.
One of the most common plants in Cyrodiil is flax, whose seeds coincidentally allow you to make powerful and light restore magicka potions once you have 100 skill. Once you reach this point, your magicka is essentially infinite. Hoard all of your Aloe Vera, Venison, Bog Beacon caps, and any flax seeds in excess of what you can turn into plain magicka restore potions, and once you hit level 50 alchemy, you can combine all 4 for a potion that restores health, restores magicka, shields you, and has a decent feather effect. Basically, immortality in a bottle. Alchemy scales up nicely with your level, too, since as your skill goes up, your equipment improves, and you have higher luck, your potions keep getting better and better.
Conjuration is the skill I use the least, mainly for RP reasons, but if you don't mind townsfolk asking you if you've been reanimating dead folk once you hit 70 in the skill, then conjuration is a great way to distract and damage your foes. I find destruction spells to be a better use of my mana if I just want to kill something, but they make good shields if I'm facing a crowd that could put me at risk.
As far as weapons, I use them for blocking more than for attacking, so something lightweight would be ideal. If you're going to be wearing armor of any kind (forgoing your 100% spell effectiveness) then you will want a light 1-hand weapon and a shield. Silver daggers are great for this, because if you find yourself silenced, chances are you're facing a ghost and other weapon materials (except daedric) won't be terribly effective. Silver shortswords are also nice, and I'm especially fond of Chillrend early on when your mana will be running out more.
That said, my main mage character uses either an unenchanted elven shortsword or an unenchanted silver mace, both exclusively for blocking and because I like how they look. In cases where I get silenced, I don't even try to dispel it with potions or swing my weapon, because dodging attacks when I'm vulnerable makes for a fun change of pace, imo.
My first mage was a Breton, and I didn't like it very much. It's just too easy--you're a good spellcaster, and all it takes is a Mundane ring for you to be damn near immortal, assuming you have some armor/shield rating. High elves allow you to cast more spells (100 extra mana, 150 with the mage birthsign, is HUGE early on) but you actually have some semblance of challenge as you start fighting spider daedra and swimming through lava. Or just pick whichever one looks the best to you
If you managed to read all this, I hope you got something useful out of it! I adore spellcasting and I'm always happy to see other people willing to try out that part of the game.
