When do set and legendary items start dropping?
UPDATED:
Legendaries can drop at any time, anywhere, albeit rarely. With the Loot 2.0 updates, new heroes from lvl 1-59 will get a guaranteed legendary drop for the first kill of The Butcher, Belial, Azmodan and Diablo.
Characters below level 11 are unlikely to find Legendary items, because they have a minimum level requirement. Currently there are no Legendary items available below level 11. (source)
For level 60 characters, you should get a guaranteed legendary drop from Diablo every time you kill him after resetting all of your quests and working your way back to him.
Higher tiers of Torment supposedly include an increase in legendary drop rates, but the exact amount of the bonus has not been disclosed. It doesn't seem to be major, though, so you should pick a lower tier of Torment over a higher one if you will kill significantly faster in the lower tier.
Note, however, that some legendaries will only drop in Torment I - VI.
Original answer:
Yes, there are set and legendary items.
Many of them will be available through crafting, requiring the recipe to drop.
Others will drop randomly, but extremely rarely. This can happen at any level, as not all legendaries are high level gear.
The best legendaries will be Inferno drops.
However, it has been stated that the very best items in the game will not be the set items or legendaries. Instead, the very best will be Inferno level randomly-generated rares.
According to this and a post from Bashiok, set items are legendaries, with the special exception that they do not drop until level 60.
Bashiok: Set items are now legendaries defined by a set bonus. They’re crafted items while leveling (recipes can be found that grant the ability to craft all the items in a set), which allows someone to really invest in filling out a set before they out-level it. Then at max level we’ll drop set pieces normally.
Before level 60, legendary items with a set bonus can be crafted if you have the recipe and a sufficiently leveled blacksmith. A list of legendary items with set bonuses is available here. Note the lowest level items listed require level 20.