First, decide what sort of animal book you want to write. Do you want to write fiction or non-fiction? Is your book for babies, young children, teens, or adults? Next, check out books that are similar to the one you're thinking of writing. Take note of publishers whose books you like.
Now, start writing your book. Even if you're writing fiction, having an outline really helps. Even if you make changes to the original plan, having a basic structure in mind helps you stay organized and on track as your story progresses. If you're writing fiction, be sure the characters are strong and the conflicts interesting. If your book is non-fiction, be sure your facts are complete and accurate, and make certain the book delivers what it aims to deliver.
Next, choose a publisher for your book. Avoid vanity publishers that charge authors a fee. Use the internet and resources like Writer's Market to identify potential publishers. If you're a new author, your best bet is likely to be a small to mid-sized independent press that accepts submissions directly from writers. Browse through the publisher's product list and envision where your book might fit in their current line. Select three or four of your top choices, and put together your submission package.
With any sort of book submission, you'll need a cover letter. This brief, one-page letter introduces you and your book to the editor, and explains succinctly why your book is needed, why people will buy it, and why you're the perfect person to write it. Send this along with your complete manuscript, or with a few sample chapters—just follow your particular publisher's manuscript submission guidelines exactly. If your animal book is a picture book, don't worry about illustrations unless you're a professional artist. Most publishers prefer to choose their illustrators independently, with or without author approval.
Make sure your proposal is professional and polished, free of typos and grammatical errors, well-organized, and well-presented. Don't use weird fonts, colors, or any other unnecessary embellishments, but do double space the text. Send your proposal package to a specific editor if possible—you may need to do some extra poking around on the internet to find out this information, but doing so will give your animal book a much better chance of actually landing in the right editor's hands. Mail your submission using regular first-class mail—don't use a mailing service that will require the people at the publishing house to sign for it. Publishers are busy, and you don't want your animal book proposal to be a source of annoyance!
Follow these tips, trust your creativity, and you have a good chance of getting your animal book published! Happy writing!