How do you get a children’s book published?

Whenever I do public events for my books, I keep finding out there are a lot of aspiring children’s book authors out there who are looking for advice. If you are an aspiring children’s book author, here are a few tips on writing and publishing a children’s book that I’ve gleaned over the years that hopefully will be helpful.

Consider becoming a member of the Society of Children's Book Writers and Illustrators (SCBWI). They are the premier organization for published and aspiring authors, and they provide insight on the craft and business of writing children's books, as well as help you make professional connections. They also have local chapters that meet regularly in person and are beneficial to be involved with (or at least mine has been very helpful to me).
A surprising number of aspiring writers have told me they have never actually written anything and ask me how to get started. The secret to getting started writing is to sit down and start writing. Yes, it is probably going to be really bad at first, but the good news is you can keep editing it until it becomes what you hoped it would be.
If you are writing a picture book, always keep in mind that it is a picture book. Pictures are 50% (or even more) of the equation. Make sure you are writing something that needs visuals to tell the story and don’t describe things the illustrations will show.
Don't try to find your own illustrator (unless you plan to self-publish). With only one exception that I know of, publishers do not want writers to submit an illustrated book that was illustrated by someone else (they love writer/illustrators, though [i.e. the writer and illustrator are the same person]). Just send your manuscript. If they like it, they will set you up with an illustrator who is likely much better than one you might find through your social circles.
I’ve met a lot of people who want to write a children’s book about their dog. As someone who loves his pets dearly, I am also aware that no one else finds my cat and dog as fascinating and entertaining as I do, so your pet is probably not the picture book topic that is going to get your foot in the door.
To find publishers who might be interested in the types of books you want to publish, go to the library and look for books with similar content and a similar approach to what you're thinking, and plan on the publishers of those books being your targets. In other words, don't submit a nonfiction nature picture book manuscript to a publisher who only does YA romance novels.
If you want to get published by one of the big publishers (or any of their imprints) like Penguin Random House or Harper Collins, you pretty much have to have an agent to get a submission in front of them. There are lots of smaller publishers who don't require agents, though.
You can find lists of potential agents and publishers through SCBWI or on the QueryTracker or Manuscript Wishlist websites.
Any publisher or agent you are interested in should have submission guidelines on their website. Follow the submission guidelines exactly and don't be pushy or gimmicky with your submission. Professionalism goes a long way. (When I was figuring out what to do and what not to do when submitting, this oldish blog was very useful to me, and also funny).
Legitimate publishers pay you to publish your book. If a publisher is charging you to publish your book, they’re basically scamming you, so avoid them.
Along those same lines, be wary of anyone who has never been published before charging you money to tell you how to get published. Unfortunately, as with all creative endeavors, there are people out there who feign expertise to prey on other people’s dreams for their own profit.
Even if you're not published yet, make a website for yourself as an author. Publishers like it if you seem marketing savvy.
As you begin submitting, prepare to be rejected. A lot. There’s a ridiculous amount of competition out there. A publisher may be going through thousands of manuscripts they received over the past few months to choose just the next twenty books they are going to publish. Those are pretty low odds. Unfortunately, when you find out you were rejected (assuming you’re lucky enough to receive verbal confirmation of this), you will have no idea if you were the editors’ 21st favorite book or 2,856th favorite book. But hopefully you are the former. Keep submitting to other places and maybe the next agent or editor (or the 35th one) will determine your manuscript is their 2nd favorite book of this publishing season and send you a contract. If you keep getting no’s, eventually even submit to the same publishers again (since there is often staff turnover and ultimately this is all subjective, plus there are timing issues and all sorts of other factors you have no control over that may change). But if you are persistent (though not pushy!) and patient and keep submitting, eventually you will hopefully get the manuscript in the hands of the right agent or editor who realizes its potential and says yes. So keep trying!