Different professions suit different stages of an Enchanter's professional career. As you gain access to better formulae you will require better items to disenchant to get the higher grade reagents {shards}.

Items made with Tailoring will garner reagents for low to mid level enchantments.
Items made with Blacksmithing and leatherworking will garner reagents for mid to higher level enchantments.
Items made with Engineering will garner reagents for higher level enchantments.

Of course you don't have to have a profession to work with your Enchanting if you don't want to. Each of those professions are intensive in themselves and it's a lot of extra effort to go to in order to be able to disenchant an item. You may prefer to just trade with other players for the items to disenchant. You may wish to have a gathering skill and sell the ore bars/gems or herbs to make money to buy the reagents or weapons to disenchant from other players or trade those raw materials with Smiths and Alchemists for items to disenchant. There's no hard and fast rule to it, it boils down to how much time and effort you are prepared to go to.

Any really as you can enchant cloth, leather or mail and plate items. If you like to throw a bit of role playing into the mix one of the more magically minded classes such as Mage, Priest, Warlock, Shaman or Druid would suit this profession best.
Is there anything else I should know about Enchanting?

High-Level Training – To train to Artisan you'll need to seek out the Master Trainer, Annora. Easier said than done because she's in the Uldaman instance in the Badlands so you have to fight your way to her to train. Be sure to take lots of money with you so you can learn as many recipes from her as possible.

Claim on Items – When playing in a group realise that you don't have first dibs on every magical item just because you need it for your profession. If your party has been set to roll on items it's not polite or fair game to take a hissy fit if others roll on an item they would like. You'll run out of parties to play with eventually as there's not much joy playing with a spoilt brat.

Magical Oils – These were introduced in 1.9 patch and allow an Enchanter to turn some of their dusts and shards into magical oils. These oils when applied to weapons add a temporary bonus to magical damage or mana regeneration that stacks with permanent enchantments. The lower levels of these recipes are found on normal enchanting vendors. The higher level versions are found on special vendors and are rewards for high Zandalar faction.

Enchanting is such a huge money sink because whereas other players are selling their magical items for reasonable profits you have been disenchanting yours and unless you start to sell your services you'll find it extremely hard to afford your mount {80 gold} come level 40.

You can make money by selling the reagents to other enchanters or by selling your services as an Enchanter to other players. What you can sell either of these for can vary from server to server but what you must remember if anyone grumbles at your prices is that the reagents you use can only come from disenchanting {some low level ones at vendors} and it's this unique ability that must be worked into your sale price.

If they bring you the reagents {which they can do by purchasing them from other players} then you will need another price list for that. So, five ways of making money and four price lists needed:

Selling reagents {dusts, essences, shards} to other players
Selling Enchantment and providing the reagents
Selling Enchantment with buyer providing reagents
Selling Enchantment and providing reagents and original item
Selling magical oils to other players

You can watch to see what other Enchanters are selling their wares for to get an idea of price structure before making up your own price list. Certainly in the first instance it's worth making a list of the items you've disenchanted and the price a vendor is willing to buy them from you for. Whatever you do never sell an enchantment for less than the total of all the items you disenchanted to get the reagents.

You are also able to enchant gloves to benefit other trade skills and people with those possessions may pay a good price to have their gloves imbued with such properties. Those available are as follows.

Enchant Gloves with To Give
Herbalism +2 to Herbalism
Advanced Herbalism +5 to Herbalism
Skinning +5 to Skinning
Fishing +2 to Fishing
Mining +2 to Herbalism
Advanced Mining +5 to Mining

You can unlearn professions for free any number of times. Go to your skills panel (press K). Click on the Profession you wish to unlearn and a small icon will appear next to it and clicking on this will unlearn the skill. Be aware you will lose all associated recipes and support skills too and if you wish you hadn't unlearnt it (eek!) you will have to start your training again at Apprentice level and work you way up again. Click the image on the right for a fully captioned shot. Use this feature with caution, Blizzard will not undo a mistake of this nature.

You can:
Enchant other players items (including soulbound) by using the trade interface.
Trade dusts, essences and shards with other Enchanters
Enchant weapons, gloves, boots, cloaks, and chest pieces.
Have one imbue {temporary augmentation from some other means} and one enchantment {permanent augmentation from Enchanting or some other means} on a single item.
Re-enchant the same item multiple times to increase your skill level.
Keep enchantments on items when you die as they are permanent

You can't:
Disenchant other players soulbound items
Sell dusts, essences or shards to vendors
Have more than one enchantment on a single item.
Sell items with Enchantments for more money at vendors.
Reverse a disenchantment
Remove an Enchantment, it needs to be replaced by another permanent augmentation.

Patch changes to Enchanting

Patch 1.9.0 (3 January 2006)
Epic items that are level 51 and above will now disenchant into a Nexus Crystal. Also items that normally disenchant into Large Brilliant Shards have a very small chance of disenchanting into a Nexus Crystal. New more powerful enchanting recipes that require these Crystals can be found in the new Ahn'Qiraj content.
Enchanters can now turn some of their dusts and shards into magical oils. These oils when applied to weapons add a temporary bonus to magical damage or mana regeneration that stacks with permanent enchantments. The lower levels of these recipes are found on normal enchanting vendors. The higher level versions are found on special vendors and are rewards for high Zandalar faction.
The Enchanting Trainer in Cenarion Hold in Silithus sells some of both new types of enchanting formulae.

Patch 1.8.0 (11 October 2005)
Enchanter-made wands are now Good (green) quality and have had their damage increased.

Patch 1.7.0 (13 September 2005)
Timbermaw Reputation Recipes are no longer sold in limited quantities with a restock time. If you have the reputation, you will always be able to buy the recipe.
Enchanting an item with an enchantment that will cause the item to become soulbound will now prompt the enchanter with a warning message before the enchantment is cast.

Patch 1.6.0 (12 July 2005)
Additional Enchanting Recipes added to Timbermaw at Honored Reputation.
Additional Enchanting Recipes added to Argent Dawn at Honored and Revered Reputation.
Lokhtos Darkbargainer in Blackrock Depths now has many additional recipes that can be learned at Honored, Revered and Exalted reputation.

Patch 1.2.0 – 17 December 2004
Enchant 2H Weapon with Intellect is now correctly giving +3 to Intellect.
Enchanter's Cowl minimum level to use set to 28 instead of 38, as was originally intended.

Patch 1.1.0 7 November 2004
Enchanting is now complete, you can perform enchantments that require up to 300 enchanting skill. Enchanting recipes generally require more reagents for 200–250 skill items
Living Essence and Essence of Undeath drops decreased.

Beta Patch 0.11.0 – 28 September 2004
All equipped items made from professions have had their stats tweaked to make them more desirable.

Beta Patch 0.10.0 – 19 September 2004
Enchanters can now enchant leather and thorium (needed for some high-end blacksmithing recipes and soon other high-end recipes in other professions).
Stats on the Smoking Heart of the Mountain changed.
Spears cannot be enchanted with 2H Weapon-only enchantments.
Fist Weapons can now be enchanted (bug fixed).
Old enchanting reagents removed from vendors (they should have been removed last push).

Beta Patch 0.9.0 – 15 August 2004
Enchanting is completely revised and has had recipes added up to 200 enchanting skill.
Enchanting now requires the disenchanting of magical items to get the required components.
There are around 100 enchanting recipes.
You can now enchant soulbound items through the player to player trade interface.

Beta Patch 0.8.0 – 7th July 2004
Heart of Fire are easier to find in this patch.