What is an easy and fast way to level up the enchanting skill? Thank you!

Depends on how you play but.... You should have like 50 petty/lesser soul gems on you after a while. Enchant a weapon with soul trap, use it, collect 50 souls, get the mage stone, enchant stuff.

Make about 300 iron daggers - this will also get your smithing to like 80. Then enchant all 300 daggers. This will get your enchanting to like 80. boom done.

Kongming is correct. Just smith Daggers and gather Soul Gems. One thing you missed though is you can buy filled Soul Gems from Merchants. If you have the cash that adds to the pile. The only way it would be easier then that is to console code the item. I made a post to ask that question but, it was never answered because it got buried. If someone knows the console code for a filled Grand Soul Gem and Iron daggers please post that. Thanks.

Slight Spoiler************* Don't forget that Azura's Star is back in Skyrim as well... very handy for soul trapping and enchanting.. although they seem to have removed the ability to soul trap your conjoured creatures... bummer, but the right thing to do for game play because soul trapping your own conjoured beast was a cheap move in Oblivion, that I did constantly