Collectors, dealers, and auction house staff are all potential teachers, and are usually generous with their time and knowledge. There's no way to develop an educated eye without looking at lots of real rugs and getting taught by people who already have some experience. However, there are some simple beginnings that can be made in the comfort of your favorite reading nook. The issue of whether a blue color is natural or synthetic is moot. The natural dye used for producing blues is exactly the same chemical as the synthetic one, indigo.
Indigo is even the dye of choice for blues in most contemporary fabrics, including blue jeans.
One of the interesting things about indigo is that it doesn't dye the entire thickness of most yarns, but only coats the surface. Thus, when the surface abrades, the dyed part of the yarn is gone and its natural color peeks through. In fact, they don't. If you've struggled with grass stains on kids' clothes, you might be surprised to learn that natural green dyes are fairly rare. Some very early Turkoman pieces have a natural dark green in their palette, but most of the greens that are seen on rugs made prior to the availability of synthetic dyes are produced by consecutively dying the yarn with indigo and a natural yellow.