I have over 600 hours into Skyrim but I've never tried a Mage. So I thought i would give it a try. I want to use Destruction, Conjuration and wear Robes but I'm not sure on anything else. I also don't want to abuse any skills that would make me over-powered like Enchanting.

Any info or tips would be of great help.

Okay I play as Dark Elf Mage so I'll give you some advice, whether or not it is good is entirely up to you.
So the first thing to do is pick your race. As you know each race has its own set of attributes toward certain skills. For mages the best that I see to use would have to be the bretons, the dark elves, or the high elves. However, being a mage does not mean you have to be one of these races; They just give you a head start in certain skills. The second thing you need to worry about is what types of spells you are going to use, which you already seem to have done. But here's the thing, each type of magic skill has perk that will greatly help with using other spells. For instance, I sometimes need to kill eneies while sneaking, but my blade and archery damage are really low. If you level up your illusion skill high enough, you can unlock the perk to cast spells silently. This greatly helped me in tons of mission that required a more... silent approach. Another thing you need to think about is how you are going to level your magicka, health, and stamina meters. The bad thing about being a mage is that you spend so much time increasing your magicka that you forget about your health and end up getting killed with one hit at times (Stupid Deathlord!!). However there are Alteration spells to cope with this as well as an Alteration Perk that allows you to have resistances when you are not wearing armor (Basically it counts clothes and robes as armor I think). Restoration is also extremely important to have for not only to heal yourself but your companions as well. There is even a perk that heals you to full health once a day when your health is low, and another that increases your health regeneration. If that doesn't work for you, then simply ensure that you give yourself a little health increase every once in a while. Or you can also include Alchemy and Enchanting. The most annoying aspects of being a mage is running out a magicka when I am in the middle of a fight. This ultimately results with my death... So you should either make sure you have a lot of magicka and health potions, which tends to get expensive, or increase your alchemy skill so you can make yourself potions with multiple combinations. As for enchanting, which I know you said you dont want, you can simply use it to to increase not only you magicka but its regeneration rate as well. I can't tell you how many times I had my but saved by the mere fact I was wearing robes and jewelry that gave me 62 points of extra magicka, but also increased its regen. rate by 62%. Im not invincible health-wise, but I can heal myself without the fear of running out of the good blue stuff.
So that should be about it, I think. Like I said this is only my opinion so whether or not you find it good is up to you. hope I helped at least a little!

I wrote this guide recently for pure destruction build. It may contain some tips you may find useful.
http://www.gamesas.com/forum/118-skyrim-general-discussion/

Destruction is widely thought to be the weakest of magic schools. Perhaps so, but I have been satisfied with its strengths. Its ranged weapons makes it best way to fight a dragon, in my opinion. Maybe conjuration is equally effective, or maybe it's more effective, but I prefer doing the killing on my own. Conjuration is very strong school. There are many people who love conjuration, but I find necromancy and dead thrall rather creepy and unpleasant.

Alteration is also very strong. I have not been impressed by mage armor, but I found paralysis and dragonhide to be very strong. Also, detect life and detect undead spells are very powerful tools for planning your battles and not getting surprised.

Illusion is also very very good. You have the muffle spell, a low level, rather inexpensive spell which makes you invisible to falmers. You also have invisibility spell when you level up, which makes you invisible. The ability to troll the game with calm/fury type spells is also very fun.

I don't use restoration that much, but recently I used 3 perks to get the avoid death perk. If you hate dying like I do, I think it may be worth it to you.

There are many cool and strong ways of playing a mage. Have fun.

Enchanting is not at all owerpowered in a vanilla game. Playing a mage, you will always need to match super-fast and super-precise enemy frost mages and such, who also have loads of health and minions. Enchanting is the only working shield in this game, for a mage that is. What are you going to do, when you will meet an arch-necromancer, who casts at you a shock spell of 80 points every second? Even 3 trinkets of 20% shock shield will not be enough... So, enchanting has to be abused (in a vanilla game sense). For a conjurer, restoration and alchemy can be a natural addition.

I agree that enemy magic user is the biggest threat for a mage. However, you can survive this game fine without using enchanting. You can instead rely on shopping and looting.

I recommend carrying light boots each with shock, fire, and frost resistance. it starts out at 15% and as you level up you will find stronger boots. It goes up all the way to 60%. I also recommend getting necklace with elemental resistance, for those occasions when you feel you want extra protection. Maybe a light shield may be useful in some situations, if you want to free up necklace for extra magicka or health. You can get to 85% elemental protection quite easily without making your own enchantments. Being a Nord gives you 50% frost resistance. Being a Dark Elf gives you 50% fire resistance.

I would also recommend getting good magic resistance. Being a Breton gives you 25% magic resistance. Becoming an agent of Mara gives you additional 15% resistance. Lord Stone gives you 25% resistance. You can get another 30% with the perks in alteration tree.

Atronach Stone gives you 50% chance spell absorption ability. There is a high level perk in alteration tree that gives you additional 30% chance. Very useful for fighting dragons and mages. I really like Atronach Stone and spell absorption ability a lot. If you use conjuration, you may not love Atronach Stone as much, since it makes your conjuration spells to fail half of the time.

I find Conjuration a great help to my mage. I don't use the undead so much, just don't really like them, but I do use the fire and frost minions until I can get Dremora lords to have around. I make sure I have good amounts of base Magicka and good regin of it, with both items and the Resotoration perk that boosts it, which greatly reduces the amount of times I run out of magic. When you add in decent minions, the times it happens is even less. My mage was a destruction/conjuror with Alteration as a third max minors in Illusion and restoration(silent casting and decent sneaking ability is great for choosing when you want to engage the enemies instead of letting them choose). Sneak was raised to max eventually but only perked up to the does not set off pressure plates when sneaking, which cuts down on the times I get hit with killing traps, since I sneak EVERYWHERE in a dungeon. I eventually added light armor to the mix instead of robes but that is your choice especially if you are NOT enchanting which. I feel enchanting is a good idea but it is YOUR choice not mine, so play the game as you enjoy.

Oh, my choice for mage was a high elf, which gives the racial skill of greatly increased mana regin for a time once per day, which early on before you get a huge mana supply, saved my bacon several times...well more than several.

First things first, don't ever try to rely on destruction alone, or even as your primary school of magic...at least not without thew use of exploit via enchanting and alchemy. regardless of what anyone tells you, Destruction all on it's own (vanilla) totally [censored] blows. But since your not going to be soley a destruction mage, we don't need to continue that part anymore.

Conjuration is an excellent choice for a mage, so is illusion and sneak. Personally, I never waste perks in the sneak tree as a mage, you would be better suited taking all the perks in illusion (believe me, you will need them.)

Enchanting really is a must as a mage. Enchant everything you find, collect all the soul gem you come across. Dwemer ruins are an excellent source of these.

Alchemy, although very handy for a mage at times, I don't consider a requirement...even in the face of destruction's weakness.

Thanks for all the info. I think I got about everything I need except one thing. Why Atronach over sommoning undead? Undead have a perk that adds 100+ health.