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.
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.
THIS ANSWER IS UPDATED AS OF PATCH 2.0/REAPER OF SOULS
All previous answers assumed an environment where the Auction House was a possibility for gem procurement. This is no longer the case in patch 2.0, and in order to get gems above the "Imperial" level, you must pay Covetous Shen to combine gems, as well as provide a few materials beyond simply money and gems at the highest levels.
So, is it worth it to combine gems?
YES (for Marquise and up).
In Patch 2.0, it is possible for Marquise gems to drop from level 61 and higher monsters. It is also possible for Imperial quality gems to drop from level 61 and higher monsters on Master difficulty and higher.
The gap between a Marquise gem and an Imperial gem is substantial, but gets even higher as you go further. Let's take an example of an Emerald placed in your chestpiece...
- Marquise: +62 Dexterity
- Imperial: +100 Dexterity
- Flawless Imperial: +160 Dexterity
- Royal: +220 Dexterity
- Flawless Royal: +280 Dexterity
So, given that you can achieve a boost of +180 to a particular stat by combining gems over what you can get just from drops, it is absolutely worth it.
Below Marquise you will have to spend to upgrade Covetous Shen and then further spend to update the gems. If you can afford it, go ahead. More likely by the time you go through the effort you will either outgrow the piece of equipment or drop higher level gems. It's really not worth the effort though.
I found both my first socketed item and my first gem in Act II after I
However, both @Sterno and @OrigamiRobot found gems before the activity above - so far, it seems Act II is the only real concrete limitation so far.
However, both OrigamiRobot and @Resorath confirm a similar experience (a gem dropping off the activity above). It seems that a gem will always drop from said mob. It does not seem a socketed item drop is guaranteed, though.
It's a few more quests until you find the Jewelry Artisan, Covetous Shen, but it is shortly thereafter in the same Act. You can find gems before him, and before the activity above.