Right, this is going to be long and spoiler-filled, but I hope it gives you a fair bit of information. This is WITHOUT your restrictions, so hopefully this will let you figure out what to do with said restrictions.
1. Doing the College quests will get you some of the best robes in the game. Otherwise, I'd become a vampire and nab the Ring of Erudite.
2. High Elf, if you're going offensive, since they get a sizable +50 boost to their magicka right from the get-go and their racial power lets them regenerate magicka extremely rapidly once per day. It's the race I would most recommend for higher difficulties, but Breton and Dark Elf are also good choices.
3. Alteration and Conjuration will be your primary forms of defense and support, backed by Destruction and Restoration. I wouldn't recommend Illusion for a pure mage build since I find that to be more stealth-catered.
4. Never rush into combat; survey potential danger first by looking around sharp corners (a clever way of doing this without tilting the camera angle or positioning yourself into a risky spot would be to simply cast Detect Life/Dead or using the Aura Whisper shout). Always cast flesh spells first (that is, Oakflesh, Stoneflesh, and so on), since you definitely will be OHKO'd by higher level enemies without that extra armor rating. That being said, you should almost never be too close to an enemy. Between your conjured atronachs and destruction spells, you should be able to take them out without too much incident in most cases. Either way, a mage is squishy, so you'll need to be moving around a lot. As for element of spellcraft, all three have their perks. Fire does the most damage, frost slows your enemies down, and shock hits the fastest and most accurately. It's really up to you. I'd go with fire if I were a newbie, though - most balanced of the three, and you'll probably want to focus on damage output.
5. Unfortunately, being a mage is expensive. Paralyze is a wonderful spell to have at higher levels, and I'd recommend virtually every spell save for a few. If you're concerned about dropping a lot of money on tomes, keep in mind that you'll end up with a lot of loot at the end of the day regardless of difficulty level. Alternatively, you could play the douche mage and steal things and sell them to the Thieves Guild for coin.
6. The College of Winterhold questline can be done in almost its entirety. Labyrinithian is probably better for level 25+, though, since you face a Dragon Priest at the end of it.
7. The only quest I could think of you avoiding is the Dawnbreaker quest, since the mage at the end of that one is an absolute asshat and extremely overpowered. But really, with the proper perks and training, as well as potions and any enhancing items that you have on hand, you should be able to tackle virtually any threat. Also, you're a mage - no reason for you to use a sword.
A typical pure mage setup that I like to run (with great results) is usually something like this:
Race: High Elf
Standing Stone: Mage, then Atronach/Lord (use Aetherial Crown). If you don't have Dawnguard, stick with Atronach - the regen nerf is easily fixable.
Skill Spread (Level 50)
Destruction: Novice-Master, Dual Casting, Impact, Augmented Flames* (2/2), Intense Flames*
Alteration: Novice-Master, Mage Armor (3/3), Magic Resistance (3/3), Stability
Conjuration: Novice-Master, Summoner (2/2), Atromancy**, Elemental Potency**, Twin Souls
Restoration: Novice-Adept, Regeneration, Ward Absorb, Recovery (2/2), Avoid Death
Enchanting: Enchanter (5/5), Insightful Enchanter, Corpus Enchanter, Extra Effect
*Can be replaced with their frost or lightning equivalents if you prefer, but I'd only pick one element to start.
** Can be swapped with necromancy perks if you prefer, but I find atromancy easier.
Obviously, my guide isn't really a guide - more of an account of my own experiences. Hope this helps, and I also hope it isn't too daunting. Part of playing a character is rooted in mixing and matching and experimenting with different skills to determine which build you're most comfortable with playing!