Brainstorm an idea that inspires you and stirs passion. Check out some children's books from the local library and note what you like about them and what might be missing. Think about how to contribute an original theme and characters to the world of children's literature. Write out a basic overview of your plot. Once you have let the ideas sink in, sit down and write. Stick with it until you've completed your book
Join the Society for Children's Book Writers and Illustrators (SCBWI) (see Resources below). This is great way to receive feedback on your story, find out about conferences where you can shop your story around and learn the ins and outs of the business. The SCBWI charges a nominal fee for membership but provide invaluable advice, connections and support for a struggling new author.
Collect a list of publishers that accept unsolicited queries regarding new material. You can find a list of these at the SCBWI's website as well as in publications like the "Children's Writer's and Illustrator's Market." Send out a letter to each of these companies detailing your book, a one sentence synopsis of the plot and your contact information.
Send out a sample chapter of your story if there is a positive response to your query letter. Pick the chapter that you think contains the most action, drama or excitement. Include a list of major characters, their descriptions and their involvement in the plot of your story. If your children's book is less than 30 pages, you can send the entire manuscript.
Get an agent. Once you have the interest of a publisher you can secure an agent quickly. Agents can help you understand your contract, negotiate compensation effectively and watch out for your best interests. The "Children's Writer's and Illustrator's Market" has a list of reputable agents you can approach. You should also check the SCBWI website for agent information. Once you have an agent, they will be able to approach publishers that do not accept unsolicited materials for your next project which can result in larger distribution for your book and higher pay.