What gem qualities are there? What can be found and what has to be crafted?

In Diablo 3 there seem to be a lot more gem quality levels than in Diablo 2.

How many quality levels are there? Can all of the quality levels be found and if so is there a breakdown on how difficulty level affects it?

If they can't then how many lesser gems does it take to create the upper tier ones?

There are a total of 14 tiers of gems. 13 that you'll learn from the Jeweler, as well as the baseline chipped gems, which are not created. You'll need 3 gems of any level to create a gem of the next level. Higher level recipes will also require an additional ingredient, one or more Pages of Jewelcrafting, Tomes of Jewelcrafting, or Tomes of Secrets. However, all the way up to Radiant Star grade gems, which require 20 Tomes of Secrets, you will still only ever need 3 Perfect Star gems to create them.

You can see all of the relevant recipes over at Blizzards item database.