Magic Skills

For the six schools of magic (Alteration, Conjuration, Destruction, Illusion, Mysticism, Restoration), the simplest way to improve the skill is to cast spells. Make a few cheapie practice spells and go to town.

The problem with this approach is that you have to cast the spell successfully and at low skill levels you have a decent chance of spell failure. It might be worth the investment of a few hundred golds per skill to train them up to where your chances of spell failure are minimal.

Sitting out in the countryside and lobbing fireballs at trees works very well if you have the time to spare. Use Hearth Heal instead of potions or sleeping. Make a ranged "Light" spell and toss it around. It's not an offensive spell, so you can even nail innocent bystanders without consequence. Use Water Walking spells instead of potions or items. Buy a "Mark" spell and cast it until you're out of magicka, rest and repeat.

Alchemy is a different matter. You only get skill points for successfully making potions or eating ingredients and getting the effect. I have heard one suggestion about camping out in the Alchemist's shop in Balmora, buying cheap ingredients, mixing them on the spot and then selling the resulting potion as a means to develop both Alchemy and Mercantile skills. Aside from that, just eat every single loaf of Bread and Small Kwama Egg that you come across.

UPDATE (August 2010): at low skill levels you have a decent chance of wasting your ingredients when making potions. Fortify your Intelligence with spell/potion/item(it only needs to last until you can open your inventory) to increase your chance of success.

Enchant is the most difficult. You get skill improvement points for making enchanted items and recharging magic items. The problem with this is that it takes filled soul gems to do either one. You'll find lots of Petty and Lesser Soul Gems, so invest in a Soul Trap spell (good for Mysticism) and soul trap those random wilderness critters and use them to pump a few charges back into your items. Alternatively, you get 0.1 skill improvement points for each successful use of a magic item, so you can combine the two by using an item until it's drained and then recharging it with Rats, Scribs, Kwama Foragers, etc.

UPDATE (August 2010): the same problem as with Alchemy - lots of failures at low skill levels and Soul Gems aren't cheap. Same solution: fortify before trying.