Here's the skinny:
You can go about this one of two ways.
1) Create a mock-up of your book, then make copies for submission (color if illustrated)
2) Find a literary agent
Search the web for children's literary agencies (verify it's a legitimate agency on WritersBeware.com), scroll through their list of agents, find out which ones are accepting picture books, send them a letter or email (it's called a query) according to their specifications (every agent is different).
Send letters & emails to as many agents as you can (I used to submit in packets of 10) because 99% of them will email or write back with a blanket 'no.' It's a complete and utter craps shoot. Eventually, after 100 or so letters, you will pass upon the agent who wants your type of book, with your type of story, and they will ask you to send them a mock-up of your book. Never send them the original!!!! You
will not get it back, and you will not get your copy back either.
They will write or email back and say yes or no. Most will say no. Again, craps shoot. But keep at it, eventually, you'll find the right agent for you. Never, ever, ever, EVER pay an agent any up-front fee. They should read it for free! No editing fee, no "copy" fee, nada, zip, zilch. They get paid after they sell your book, they take 15% of all sales.
3) Agent notes
The agent will have "notes" on the book (or, they won't have any) and it is 100% up to you if you think the changes they suggest help or hinder your book.
The agent will submit your book to publishers. You will hear nothing for a long time. Call and check in with your agent at least once every few weeks so they remember you exist. Technically, they work FOR YOU, so don't let them treat you like they are doing you a favor.
a) The publisher will make you an offer
b) Your agent will argue for you to get you the best deal
c) It will include an advance (anywhere from $1,000 on the low end, to $10,000 on the high end for children's books). You will receive (standard deal) 10% of hardcover sales and 5% of paperback. For e-pub, it's a bit better, some traditional publishers offer a 25 to 30% royalty.
d) The publisher will have notes on your manuscript. These are harder to argue. Make the changes you feel are necessary, but you don't have to make them all.
e) The publisher makes the cover (you have no choice in the matter) - sometimes with picture books they'll use a piece of the art from inside, but that's not a guarantee.
f) The book will come out about nine months to a year after the deal closes (I know, terrible!) The publisher will give you a little marketing push, but after that you are on your own. That means, blog, tweet, FB, and reviews are almost all up to you.
Warning: You will not get rich selling only 1 children's book. You may be able to make a living if you publish 10.
1) Create mock-up and make copies
2) Research publisher submission guidelines
Some publishers will not accept "unsolicited" submissions (submissions they didn't ask for, or ones that did not come from an agent), but some will. This will take a bit of ground work to see which ones you can submit to, and which ones are closed. If you find one that is open and accepting picture book submissions, write a query as per their specifications and wait for a response.
3) If they accept your book you will have to negotiate your advance & contract by yourself
(Personally, I do not recommend this! But I've known some authors who have.)
4) See #5 Above
Please feel free to ask as many questions as you like.
The reason I recommend the SCBWI is because they have periodic "seminars" and "conferences" where agents, editors from publishers, and other published authors give speeches and advice. Some editors and agents will even accept submissions from attendees of these seminars, and it does carry clout among them if you write in your query you are an SCBWI member. Also, the SCBWI holds contests and if you win one of those, you also win submissions to agents and editors as well.
Hope this helps!