Question: I have written a children's book that is being illustrated by a friend. What do you think is the best way to publish my book?
Answer: Does your illustrator friend work for a publisher? Be aware - many publishers prefer to use their own illustrators. If your friend has illustrated a couple of books, I would ask her for her contacts. Since you've yet to publish a book, if I were in your shoes, I would utilize her popularity to get your foot in the door. This industry is difficult to enter.
If you do a search online for children's book publishers, there are lots who are clear about what kind of stories they are interested in. You might want to look up books you feel are similar (animal main character, saving the day, etc.) and see who published their books. Each publisher has their own guidelines for submission, so make sure you follow those exactly. (A writer who can't follow submission instructions turns off publishers - they assume the writer will be difficult later.)