Here is a more detailed "Do and Don't" list I think is important to follow:
* Believe in yourself and your art. If you don't like it people won't either. Trust what you do and try your best at it.
* Practice, practice and did I say.. practice? Draw a lot. From real life, from references in books, from your imagination.
* Try to find a style. This is not easy, I know. I'm sure people say it all the time like everybody knows a magic formula as to how to get one. There is no formula, but what I think is the best way to approach it is by practicing so much and closing your eyes and thinking what it is you REALLY want to draw. Is it animals? Kids? Pirates? Then do it... and after a while a pattern will appear. You'll see some similarities to them and that is the style emerging. Also I think that style is linked to what you enjoy the most doing.. Usually you have more confidence and a looser hand at that so the style will come easier through this.
* Read all you can about it. The art side and the business side too. Buy books and read articles in the internet.
* Subscribe to the SCBWI. It is a great way to learn about this business. Read the forums and ask questions. Try to attend conferences, you get to meet many interesting people in the field and learn from all the wonderful speakers.
* Work on a portfolio. When you "find" your style, work on having 10 to 15 really good pieces for your portfolio. Since this is a Children's Book Portfolio try to include children in it. No still life or anything like that. Try to tell a story with your illustrations and always include whatever it is that you want to be drawing because ultimately, you will be hired to do what ever to have in your portfolio.
* Take advantage of the internet. This is an amazing way to learn, meet people and be seen! I can't tell you the importance of this. Open a blog, make a website, take advantage of portfolio sites (free: Coroflot, CreativeShake -Paid: ChildrensIllustrators, Picturebook etc... ) and post your work there.
* When you have done all from above, then research in bookstores, libraries and the Children's Market Guide for publishers that meet your style. Look for books that speak to you, that you think have similar feeling or style as yours then find out what the submission requirements are for them and then send out samples of your work. Now don't SIT and wait after this because it will be a LOOOOONG wait. Instead keep reading, learning, drawing, making new pieces for your portfolio, blogging, practicing, experimenting and dreaming... never stop dreaming. I know this is what keeps me going.
* Don't skip ANY of the important steps in the "Do" section! Just do it!