I'll point this out now that Alchemy very early on can cause you to fly up levels if your not to careful and you will have to watch this unless you want to get over powered to early on. Zak will disagree with me on taking Alchemy out if you are new to the game, Although I know how strong it can be I choose to take it out of my major skills and swap it for one that wouldn't level as quick. Hence I took Mercentile. So I'll talk you through my custom Magi class that is the same as the basic Mage class except for the swap of Alchemy for Mercentile,
I don't see the problem that people are having with Alchemy as a major. It seems to me that the advantages outweigh the disadvantages, for a mage.
If you're playing a Breton, Argonian, or Altmer with Magic as a specialization, Alchemy starts as a major at 40. You have access to two alchemy effects, and you only have to raise Alchemy by 10 points (which accounts for raising your character by only one level) in order to reach Journeyman. At that point, you have three alchemy effects for each ingredient, and can make potions like Invisibility. (Invisibiity at level 2!) From that point, you're only making the potions you actually need, so the skill only advances as fast as you want it to.
A Bosmer can actually start with an Alchemy of 45, and thus can have journeyman potions available at level 1.
And even if you were to spam Alchemy at early levels, it's only going to push your character up by five or six levels, which is hardly enough to imbalance the game if you're playing as a mage.
Conjuration - Summon powerful creatures to your side to help you out. I use this school an awful lot and it was the first of the six magic skills that I mastered.
Conjured creatures are good distractions, but are poor substitutes for your own offensive skill unless you are playing at a very high difficulty setting. Conjuration's Bound armor and weapons are an under-appreciated feature of the skill, I'd say.
Illusion - Frenzy on the right enemy can cause a lot of trouble causing it to rip its friends apart. Although most of the time you will encounter enemies on a solo basis Illusion can still be fun. Invisabilty on yourself along with Nighteye can allow you to sneak around dungeons or sneak up on an enemy. I mainly use Illusion magic to sneak around or run away from a fight really quickly.
Illusion works that way, but it's also one of the most powerful offensive skills in the game. Command (which is more effective at handling groups than Frenzy) is a powerful offensive weapon. Frenzy, in my opinion, is more useful at getting an NPC to attack you, so that the guards and other NPCs will be on your side.
Restoration - You would think it would be pure healing spells? Restoration at 50 and you can start to cast fortify spells. These will increase your base stats for a short period of time or you could use it to raise your health pool if you think that the next fight is going to be hard. You can also use Restoration magic as an Offensive Magic school if you choose to do so. Drain Health is a powerful spell when used the right way.
Drain Health is a Destruction magic. I think you're referring to Absorb Health, which is indeed an excellent offensive weapon (it only works at short range, so it's an On Touch skill.)