Playing a bit on the PTR, I've been able to try out some of the new Transmog system. Here's what I've discovered:
You still need to go to a Transmog NPC in order to actually change your appearance, and this will still cost gold. Also, hiding helmets, shoulders, and cloaks will now be handled by your transmogrifier, rather than as an option in the interface preferences.
You'll immediately have access to anything you can and should equip in your inventory, bank, and void storage (should meaning if you're a warrior, it's plate,) as well as any quest rewards you may have gotten over the years. I was actually a little surprised at how few, say, helmets I'd gotten, but you also have to remember that vanilla quest rewards were far less frequent and regular than they are now. You couldn't always expect to get a full set of gear from doing all the quests in a zone.
Weapon enchantments can also now be transmogrified, similar to the weapon illusions you could get with a follower in the enchanting hut in the garrison.
It appears that getting these enchantment appearances requires you to get a tome of enchantment appearances, which are made by enchanters.
While you can only change your appearance at the usual NPCs, you can look at the appearances you've collected and set favorites, as well as even specifying which exact item you're transmogging - like if you want it to be clear that this is a Hammer of the Naaru and not the Sunsoul Warmaul, even though the two are identical in looks.
You can also save appearances for specific specs, which I think means that switching specs will also change the look of your armor, even though, given how the new stats work, you'll basically never have to change armor between specs. This needs a bit more testing on my part.