Should I stock up on a bunch of grand soul gems, fill them up, and start enchanting, or reuse Azura's star over and over? Do different enchantments give you more exp for enchantment? And do the levels of the souls increase it faster, like a grand soul would give you more exp than a greater soul?

The size of the soul gem and the enchantment has no effect of the skill gain. What I do is I buy a bunch of soul gems then fill them all. Then I make a bunch of iron daggers and enchant them. Then I sell the enchanted daggers and buy more soul gems.

Force856 nailed it. It takes time, lots and lots of time.

Thank god I saved every single one of my soul gems instead of selling them. I used up all my petty soul gems (the ones that were filled up), and finished it off with some lesser soul gems. Didn't take too long to get from 93-100 by doing that.

Higher the value of the created object, faster you level up. Enchant everything with banish and you'll get there pretty quickly.