Before beginning any mage build, head to the college of winterhold to learn what spell is required for your test. Because a mere thirty gold is all that is reqired, your can buy some spells much cheaper than usual.
For levelling conjuration: Head to Whiterun and buy soul trap from the mage in the Jarl's palace. Just inside Whiterun is a temple to Arkay, whose priest is in need of your help. They've lost their amulet, source of their diving powers, inside the catacombs, and the dead are becoming restless. Enter the catacombs and take a left towards the door. Kill one or two of these naturally occuring undead and cast soul trap on the remains until you run out of magic. There is a third skeleton, but as long as they don't see you, you should be able to wait one hour and repeatedly cast soul trap again. Doing this should allow you to level up until you can cast any spell you need to for your test.
For leveling restoration: In riverwood, there is a blacksmith's forge that you can easily jump up and down on top of. If you don't want to do so in a public place, there should be a iron mine near the guardian stones. The mine is being used as a base of operations by the bandits who went as far as building a secret forge inside of it. Although the bug that cause destruction to level up was fixed, you should still be able to hurt and restore yourself in the flames. With only the healing spell you have by default, you should be able to get Novice Restorat, Dual Casting, and the first Recovery perk without much trouble. With these perks activated, you should be able to recharge the magical energy you use healing yourself while jumping up and down.
For leveling alteration: recommend using pick pocket for the first few levels. Head to windhelm, and find the unseen beggar woman. She will give you lessons, but only take one lesson at a time. After one lesson, close the dialogue and either save the game or enter a nearby building to autosave. Then try to pickpocket your tuition using what you have learned. Load your last save if you get caught, but you should be able to steal the gold from one lesson. Buy some perks and it should soon be almost too easy. Then head to the mages college and talk to old Tolfir. If you haven't completed Saarthal, he's very easy to find. Using the same one lesson strategy, you can level up for free. However, it becomes almost impossible to steal tuition money after level 51, when the price suddenly jumps. By that point you should be able to buy Detect Life spell. Buy it and head to the woodchopping block in front of the Bannered Mare Inn in Whiterun. The marketplace is full of people during the day, and the inn is full of people drinking at night. If you have a woodcutting ax, your magic meter will recharge while you are chopping wood, which you can sell to the innkeeper. Even with the jump in price, you can still afford a few lessons if you are too impatient to grind using this method. However, you only need to grind up to 75 Alteration "Exper Level" when Tolfir sells a paralyzation spell. When a person is hit with this spell, they will freeze and fall to the ground and you cannot interact with them while frozen, but when they start to get up, you can pickpocket any amount of gold from them as part of an exploitable glitch. Thus you can get free lessons from any skill afterward, all the way to 100. Unfortunately Dragons seem to resist the paralyzation spell, or it would be fun to watch them fall out of the sky.
To Train Illusion Magic: Purchasing the Muffle spell and repeatedly casting it is makes levelling Illusion Magic so easy that I actually use it to advance to the next level after buying five lessons from a skill trainer. Also the Illusion trainer gives you Mystic Tuning gloves as part of a quest. Once you clear the first of these focal points, your magic meter should suddenly start refilling incredilbly fast, as though you became attuned to all the excess energy. If you want to be a pure illusionist, simply cast muffle and walk around the first focal point until the effect wears off and then clean up the other focal points and go back to the trainer for your higher level illusion spells and the quest for the ultimate illusion spells. However, like most videogames, "bosses" or high level encounters will have an immunity to "a simple illusion."
Only Conjuration and Destruction seem to be of any use in combat. The damage cap on the continuous spells and the charge time of the bolts seems to nerf Destruction, but spells from that school can be used in a staff form while the mana is reserved for restoration, etc.