Beginning
Again, the first advice I would give is to go to a bookstore or library. There are many books that give step-by-step advice for artists and writers on how to become a published author or illustrator. Try the annual editions of The Writer's Market or The Artist's Market.
Classes can be very helpful.

Professional Artists
If you are a professional artist, I would advise joining the Graphic Artists' Guild. The Guild Handbook is invaluable and provides an exceptional guide to pricing and ethics. You can order it through the Guild.

Research your market and learn about publishing companies and their policies. Submit samples to the companies you feel are the best match to your work. Most will want to see a portfolio of your work. Seek out companies with a reputation for treating artist's fairly and respecting artists rights. Also, you may want to seek the advice of an attorney when dealing with contracts. It is pricey but their good advice will pay off in the long run.

Although it is difficult, I recommend that you always maintain the copyright on your work.

I do not work with a rep or an agent, but many artists feel this is the best way. Agents generally take 25-35% of your payment, but can also take a lot of the headaches away.

I feel that my success is due to a lot of things, but it didn't happen overnight. Be persistent, believe in yourself, and follow your passion! I sincerely hope this advice will help you with your journey.