Can you un-enchant an item or re-enchant on top of one to put a better enchant on it?

I know when you unchant something it destroys the item, but with a certain perk (cant think of name) you can add multiple enchantments to it

You can't add another enchant to the item that's already enchanted, even with final perk in Enchanting tree. The reason why is explained in one of the Enchanting skill books that can be around Skyrim. The only way to have two enchants on an item is to put them both on at the same time (if you have the perk).

Ahh, that's a bummer. So I'll have to craft a whole new set of armor when my enchant skill gets up there.

That is a bummer.