Research a particular field deeply. Professional appraisers for Christie's often have advanced degrees in art history or a related field--some have master's degrees but many appraisers hold doctorates. The advanced training that goes into achieving a graduate degree helps appraisers ascertain the value of a particular work.
For example, if you're looking to appraise a drawing by Ingres, you should know where the drawing stands with Ingres' oeuvre (which means an artist's total body of work). You should know how Ingres' drawings compare to other works around the same time and understand Ingres' standing within the art world today.
It helps to have an understanding of how much other Ingres' drawings have sold for within recent years--there are free websites like Artnet that can assist with such research--however the best databases of auction sales are available by subscription only.
Be a connoisseur, a term for people who have a great deal of knowledge about the fine arts, and are particularly experts in matters of taste. As Erwin Panofsky wrote in his classic "Meaning in the Visual Arts" from 1955, "The connoisseur might be defined as a laconic art historian, and the art historian as a loquacious connoisseur."
Develop your eye; that is, look at a lot of art. The more you look at art, the more you will understand what makes an artwork good as opposed to bad. For example, while Thomas Kinkaid and Albert Bierstadt painted similar subject matters--awe-inspriring American landscapes--there are crucial distinctions in technique and style which make Bierstadt's work considerably more valuable than Kinkaid's.
Understand differences in medium. For example, prints and photographs, though valuable, are considerably cheaper than paintings. Photographs and prints are reproducible--that is, the artist makes a single negative or printing plate and can produce multiple copies of the same image. Oil painting, on the other hand, cannot be reproduced in the same media, hence paintings are much more expensive.
Christie's often has sales devoted to particular media, like "Twentieth Century Prints." If you browse through a couple of these, you should begin to have an idea of the pricing differences between artists and media.