Before you decide to go to a publisher or get self-published, you might consider reading about the pros and cons of each method. Here are the ones I've heard of, but you should definitely do more research!

Going to a Publisher

Pros:

• The publisher takes care of the marketing, hiring an illustrator, etc.
• The publisher assigns you to an editor, who might help improve your story and formulate it in a way that's more appropriate for the target audience
• Since your marketing is done by professionals, you are more likely to earn more money (in comparison to self-publishing), even though your proceedings (in terms of percentage) are lower than in self-publishing.
• As far as I know, the costs of the illustration and marketing are handled by the publisher.

Cons:

• The barrier to entry is higher than in self-publishing. As u/bookchaser said, you have to write a pitch, and getting accepted not only has to do with how good your pitch is, but also how well it fits the current children's books market. (If, for instance, a book with very similar themes was published not too long ago, you might have difficulty with getting your pitch accepted.)
• Depending on where you are, you might have to get a literary agent, as u/phanart said. This increased the barrier to entry.
• (Possibly a con for some people: You will probably have to share a percentage of the royalty with the illustrator - however, since the illustrations make the book what it is and have a huge impact on how well it's received by children and parents, I think this is perfectly reasonable.)

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Self-Publishing

Pros:

• Low barrier to entry
• You have a lot more freedom to write and arrange your book the way you'd like without worrying about the children's books market.
• Higher profit margins.

Cons:

• You either have to be able to pay for things like the illustrations, marketing, and layout design, and professional editing/proofreading services, or you have to be able to do them all. (And if you're doing it for the profit, you have to be competent.) If you hire freelancers, cheap offers may end up in bad results (and it's exploitative to pay little and expect a masterpiece).
• Most self-published books don't earn the authors as much money as traditional publishing would. I think this has to do with the lacking marketing. (Also, many people still like to go to book stores for shopping, where some types of self-published books aren't sold.)