Enchanting is not particularly profitable, but it is needed for the Loop of Power and Ultimate Gear.
Farengar Secret-Fire
Talk to Farengar Secret-Fire at Dragonsreach in Whiterun for a crash course.
Use an Arcane Enchanter for the first time.
Check Enchanting under Help in the pause menu.
Give me your SOULS!
You get souls by trapping them yourself or finding pre-filled gems.
Buy whatever filled soul gems you can afford, particularly the cheap ones and the Grand ones. Cheap souls are perfect for learning Enchanting, as the size of the soul does not matter. Save the Grand Souls for your own stuff.
Enchant your bow with Soul Trap, duration 1 second. Kill stuff to fill your soul gems. While you could put all kinds of cool enchantments on your bow, stick with Soul Trap until you reach Enchanting 100.
Once you have a soul trapping bow, buy all the empty soul gems.
Fiery Soul Trap is strictly better than regular Soul Trap.
If a soul finds its way into a soul gem that is larger than necessary, simply drop the soul gem on the ground and pick it back up to reset it.
It's not much, but it's honest work.
An Arcane Enchanter.
A filled soul gem.
An unenchanted item. (You cannot replace or improve enchantments.)
Get as much of this as you can.
An Enchanting skill of 100, with the perks Enchanting 1-5 and Insightful Enchanter.
Ahzidal's Relics for Ahzidal's Genius.
Black Book: The Sallow Regent.
A soul gem filled with a Grand Soul.
A Potion of Fortify Enchanting.
Equip Ahzidal's Relics.
Read The Sallow Regent and choose Seeker of Sorcery.
If you are using a Potion of Fortify Enchanting, save and do a dry run before you drink the potion! The potion will wear off while you stare at the menu.
Enchant the item(s.)
(Don't forget to switch back to your armor and Seeker of Might.)
Yes, please.
Disenchant an item at an Arcane Enchanter to learn its enchantment. It is better to disenchant than to sell.
Enchantments to keep an eye out for:
Fortify Alchemy.
Fortify Alteration.
Fortify Archery.
Fortify Carry Weight.
Fortify Smithing.
Soul Trap (unless you have Fiery Soul Trap.)
Waterbreathing.
Paralyze. It'll start showing up in shops at level 22.
If you find one of these enchantments in a shop, buy it.
Ahzidal's Boots of Waterwalking in Kolbjorn Barrow
Note: Falling on water while wearing these won't hurt you.Wearing any four of Ahzidal's Relics gives you a +10 Enchanting bonus called Ahzidal's Genius.
Talk to Ralis Sedarys at Kolbjorn Barrow, south-southeast of Raven Rock on Solstheim, and complete the Unearthed quest. This quest is split into four phases with three days in between. You will get a letter when it's time to continue.
There are seven Relics in all, but two are rings and two are helmets. You'll need to reach phase four to complete a set. You don't need to defeat Ahzidal.
Enchanting
You learn Enchanting at a flat rate not affected by your choice of enchantment, item, or soul.
Set aside your Grand Souls so you don't accidentally waste them. Store them in a container or hand them to your follower.
Check if your bow needs recharging.
Optionally, activate the Mage Stone for 20% faster learning.
Optionally, sleep in an owned or rented bed for 10% faster learning.
Enchant! See Enchanting Items above to make the most of this. (Don't bother with potions.)
(Don't forget to switch back to whichever Guardian Stone helps with Archery.)
You can make a modest profit with these enchantments, in descending order.
Banish Daedra on Iron Daggers. It'll start showing up in shops at level 22.
Fortify Sneak on Leather Bracers or Gold Rings.