A tattoo artist doesn't really need to have any education past high school or even a high school degree, most places require licensing though. If I were to pursue this career I would like to have a basic art degree in order to get better at 2-D design.
A graphic designer on the other hand require a Bachelor's or an Associates degree in fine arts or graphic design; along with some basic skills like creativity, communication and problem solving. You might also want to take a couple of crash courses in computer graphics and design software.
A bachelor's degree is required for most entry-level and advanced graphic design position even though some may only require an associates.
Some of the classes you would have to take include: studio art, principles of design, computerized design, commercial graphics production, printing techniques, and web site design for design courses. You also will probably need a liberal arts education that includes: art history, writing, psychology, sociology, foreign languages and cultural studies. Marketing and business are useful in helping designers work better.
