I've read a lot of post about mages svck, mages being OP and whatnot.. But that doesn't matter, and I don't want a discussion about whether mages are good are or not.
I want to make a pure mage. So I would like some tips on what perks to choose, tips on what gear to choose, what spells to use and is it worth maxing out alchemy/enchanting? And where to go first. Should I do college of Winterhold as early as possible? Are there other places/quests that are good to do early on?

Lets get a good mage discussion going.

You can't really get far in Winterhold until around lvl 10 so don't get excited too early. Now as far as tactics? I'm a spellblade, I find it very difficult to use magic without a backup, I'd recommend at least getting and keeping a dagger.

I see. I would like not to use any weapons, but will keep one on me if I need it.

Pick up a rune spell ASAP, 50 damage is a ton early on and is worth the large mana cost it requires. conjuration is really good for keeping yourself alive and distracting the enemy. I got the flame atronarch really early and I'm still using it at level 21.

The difference between a Mage being underpowered and completely overpowered is the Impact perk. As soon as you get second level spells and impact, get as much mana regeneration and Destruction mana cost reduction as you can, and you're set. Pray you don't encounter something which can't be staggered, and keep a ton of potions on hand for when you do, but you should be able to find/buy enough without needing Alchemy, and Enchanting is nearly useless for a pure mage.

Be altmer get a ward, better healing, oakskin, a decent far and close range spell and a familiar never hurts.

Make sure to focus on your destruction perks. You'll get novice spells cast at half the magika when you start the tree, then at 40 destruction you can get your apprentice spells halved which is nice because if you want just a pure mage, those are your main weapons. Flames can do a lot of damage at earlier levels, so when you can, toss a point in that branch for 25% more damage. conjuration is nice to get (but mp heavy), as you can send summons to take some of the aggro from you while you drink potions and blast away the enemy. Lesser ward is a nice spell to have against other mages at low level.
There is a lot of strategy to use with other schools of magic, they all have their place, focusing on halving your destruction casts is priority, which should keep you going for at least the first 10 lvls.

I can't remember the names right now, but i believe and mora tapinella and red mountain flower (both ingredients you can find in most places) will give you restore magika potions when mixed, collect them when you see them and make lots of potions, you'll need them!