The answer to your question is going to depend a lot on what kind of children’s book it is. Is it a picture book? A chapter book? A middle grade? A young adult? Is it fiction or non-fiction?
Each of these will have slightly different advice to them, though the general advice is going to be the same. You have three choices:
1) Traditional New York Publishing. This is the hardest route, but also the most “legit.” To accomplish this, you’ll have to go through the slow submission process, and hope that an editor falls in love with the story.
2) Small press. Most states have small, independent presses that do limited print runs of books. It can be a little less competitive, but it will pay less, and the book won’t be in bookstores nationwide.
3) Self publish. This is more legit now than it was in the past–but it’s very hard to do for books like picture books. Generally, self publishing these days is done in ebook form. In this, you pay for your own editing/design/cover and release the book yourself, promoting it yourself. You spend money up front, but have the chance of reaping all the rewards as well.
What you should avoid are presses that are somewhere between 2 and 3. A press that will charge you up-front fees, making you pay for everything, instead of footing the bill themselves, but who will basically treat your book like a self-published book, giving it no marketing support.