According to Steven Winthrow in "Secrets of Digital Animation," the first steps of the animation process are the same for both animation types. Typically, an animated story begins with a script. From the script, storyboards are created.
Traditional animation requires cels and backgrounds to be created separately; computer animation doesn't. According to animator Jan-Eric Nyström, hand-painted backgrounds are created much like a landscape painting. The animator draws the character in relation to the background. Each frame is drawn and then transferred onto a cel, which is a thin, transparent plastic sheet. The sheets allow for a limited number of drawings to be used in multiple ways and in combination, thus cutting down the amount of drawing required. Each cel is then photographed until the film is shot.
Computer animation requires more steps. According to Winthrow, animating a film involves modeling (a wireframe representation of the character); rigging (the "bones" of the character); texturing, layout and animation (character motion and camera moves are mapped out); lighting and shading; rendering (creating a geometric "description" of the model); and compositing (combining the elements).
The principles of traditional animation are often applied to computer animation. These help make a character come alive on screen.
According to Shanna Smith on AnimationArena.com, regardless of the direction an animator takes---traditional or computer---he should plan on attending some sort of training program, because it's difficult to develop the skills required to be an animator without schooling.