Most graphic design positions will require a college degree of some sort, be it a Bachelor of Arts, a Bachelor of Fine Arts or an Associate's degree. Many colleges offer these programs, from traditional art schools to community colleges. In working towards your degree, you'll learn about basics of composition, using type and two-dimensional design.
This attribute is something that can't be taught but can be learned. As a designer, you'll need to have an intuitive sense of what looks good and what doesn't. While rules can be followed to some extent, above-average design requires a visual sense common to all talented artists and designers.
Designers never work in a vacuum -- they are part of a team working towards a common goal. Good communication skills, both written and verbal, are essential. You'll also need to be able to take constructive criticism from other team members and incorporate it into your revisions.
While you'll learn computer design skills in any college program, it's essential you be intimately familiar with the major graphic design programs, including Adobe Illustrator and Quark. Even if you create your drafts by hand, you will need to create a digital copy of your work as a final product, so you have to be as skilled with the computer as you are with a pencil.